Penn Institute for Economic Research
Department of Economics
University of Pennsylvania
3718 Locust Walk
Philadelphia, PA 19104-6297
pier@econ.upenn.edu
http://www.econ.upenn.edu/pier
PIER Working Paper 08-040
“Bounds on Revenue Distributions in Counterfactual Auctions with Reserve
Prices”
by
Xun Tang
http://ssrn.com/abstract=13119913
Bounds on Revenue Distributions in Counterfactual Auctions
with Reserve Prices
Xun Tang
Department of Economics
University of Pennsylvania
September, 2008
Abstract
In …rst-price auctions with interdependent bidder values, the distributions of private
signals and values cannot be uniquely recovered from bids in Bayesian Nash equilibria.
Non-identi…cation invalidates structural analyses that rely on the exact knowledge of model
primitives. In this paper I introduce tight, informative bounds on the distribution of revenues
in counterfactual …rst-price and second-price auctions with binding reserve prices. These
robust bounds are identi…ed from distributions of equilibrium bids in …rst-price auctions
under minimal restrictions where I allow for a¢ liated signals and both private- and commonvalue paradigms. The bounds can be used to compare auction formats and to select optimal
reserve prices. I propose consistent nonparametric estimators of the bounds. I extend the
approach to account for observed heterogeneity across auctions, as well as binding reserve
prices in the data. I use a recent data of 6,721 …rst-price auctions of U.S. municipal bonds
to estimate bounds on counterfactual revenue distributions. I then bound optimal reserve
prices for sellers with various risk attitudes.
KEYWORDS: Structural auction models, interdependent values, a¢ liated signals, partial
identi…cation, counterfactual revenue distributions, U.S. municipal bond auctions
0
This paper is a chapter of my Ph.D. dissertation at Northwestern University. I am grateful to Robert
Porter, Elie Tamer and Rosa Matzkin for numerous helpful discussions. I also thank Arnau Bages, Federico
Bugni, John Chen, Xiaohong Chen, Ivan Fau, Joel Horowitz, Yongbae Lee, Charles Manski, Aviv Nevo,
William Rogerson, Viktor Subbotin, Quang Vuong, Siyang Xiong and participants of the Econometrics and
IO seminars at Northwestern University for comments and feedbacks. Special thanks to Artyom Shneyerov
for helpful discussions and kind assistance with data acquisition. Financial support from the Center of Studies
in Industrial Organization at Northwestern University is gratefully acknowledged. Any and all errors are my
own.
2
1
Introduction
In a structural auction model, a potential bidder does not know his own valuation of the
auctioned object, but has some noisy private signal about its value. Bidders make their
decisions conditional on these signals and their knowledge of the distribution of their competitors’private signals and values. A structural approach for empirical studies of auctions
posits the distribution of bids observed can be rationalized by a joint distribution of bidder
values and signals in Bayesian Nash equilibria, and de…nes this joint distribution as the model
primitive. The objective is to extract information about this primitive from the distribution
of bids, and to use it to answer policy questions such as the choice of optimal reserve prices
or auction formats. (See Hendricks and Porter (2007) for a survey.) Depending on whether
bidders would …nd rivals’ signals informative about their own values conditional on their
own signals, an auction belongs to one of the two mutually exclusive types : private values
(P V ), and common values (CV ).1 These two types have distinct implications for revenue
distributions under a given auction format.
In this paper I propose tight, informative bounds on counterfactual revenue distributions
that can be constructed from the distribution of bids in a general class of …rst-price auctions
with interdependent values and a¢ liated signals. The counterfactual formats considered
in this paper include both …rst- and second-price auctions with reserve prices.2 Thus I
introduce a uni…ed approach of policy analyses for both P V and CV auctions that does
not require exact identi…cation of model primitives. My method is motivated by several
empirical challenges related to structural CV models. First, several policy questions have
not been addressed outside the restrictive case of P V auctions due to di¢ culties resulting
from non-identi…cation of signal and value distributions.3 For a …xed reserve price, theory
ranks expected revenue for general interdependent value auctions with a¢ liated signals, but
the magnitude of expected revenue di¤erences remains an empirical question.4 Another open
issue is the choice of optimal reserve prices in general interdependent value auctions with
a¢ liated signals and …nite number of bidders.5 Since model primitives cannot be recovered
1
I use the term "interdependent values" for a larger class of auctions that encompass both P V and CV
auctions. The formal de…nition of a P V auction is one in which bidders’values are mean independent from
rival signals conditional on their own signals.
2
In this paper, I use the term "second-price auctions" exclusively for the sealed-bid format. This does
not include the open formats, or "English auctions".
3
For a proof of non-identi…cation, see La¤ont and Vuong (1996).
4
The only exception is the case with i.i.d. signals, where expected revenue from …rst-price, second-price
and English auctions are the same regardless of value interdependence.
5
An exception is symmetric, independent private value auctions, where the optimal reserve price is
3
from equilibrium bids in CV auctions, these questions cannot be addressed as in P V auctions,
where point identi…cation of signal distributions helps exactly recover revenue distributions
in counterfactual formats.6 Second, it is di¢ cult to distinguish P V and CV auctions from
the distribution of bids alone under a given auction format, even though the two have
distinct implications in counterfactual revenue analyses. La¤ont and Vuong (1996) proved
for a given number of potential bidders, distributions of equilibrium bids in CV auctions
can always be rationalized by certain P V structures. Empirical methods that have been
proposed for distinguishing between the two types often have practical limitations. They
either rely on assumptions that may not be valid in some applications (such as exogenous
variations of number of bidders, as in Haile, Hong and Shum (2003)), or they may entail
strong data requirements (an ex post measure of bidder values as in Hendricks, Pinkse and
Porter (2003), or many bids near a binding reserve price as in Hendricks and Porter (2007)).7
Third, the empirical auction literature has not considered the magnitude of the bias if a CV
environment is analyzed with a P V model in counterfactual revenue analyses.
I propose a structural estimation method through partial identi…cation of revenue distributions to address the questions above. First, the bounds on revenue distributions are
constructed directly from the bids, and do not rely on pinpointing the underlying signal and
value distributions. Second, the bounds only require a minimum set of general restrictions
on value and signal distributions that encompass both P V and CV paradigms. Third, the
bounds are tight and sharp within the general class of …rst-price auctions. The lower bound
is the true counterfactual revenue distribution under a P V structure, while the upper bound
can be close to the truth for certain types of CV models. Hence the distance between the
bounds can be interpreted as a measure of maximum error possible when a CV structure is
analyzed as P V in counterfactual analyses. The bounds can be nonparametrically estimated
consistently. Although I do not provide point estimates of revenue distributions, the bounds
are informative for answering policy questions, as they can be used to compare auction
formats, or to bound revenue maximizing reserve prices. The analysis can be extended to
risk-averse sellers immediately given the sellers’utility functions.
identi…ed from the distribution of equilibrium bids. Levin and Smith (1994) also showed in symmetric …rstprice auctions, where signals are a¢ liated and values are interdependent through a common unobserved
component, the optimal reserve price converges to the seller’s true value as the number of potential bidders
n goes to in…nity. Yet the theory is otherwise silent about identifying optimal reserve prices with a …nite n.
6
See Guerre et.al (2000), Li, Perrigne and Vuong (2002) and Li, Perrigne and Vuong (2003) for details.
Also note in P V auctions, the distribution of signals fXi gni=1 are equivalent to the distribution of values
fVi gni=1 under the normalization E(Vi jXi = x) = x.
7
A binding reserve price is one that is high enough to have a positive probability of screening out some
bidders.
4
My paper is related to the literature on robust inference in auction models. Haile and
Tamer (2003) use incomplete econometric models to bound the optimal reserve price in
independent P V English auctions, where the equilibrium bidding assumption is replaced
with two intuitive behavior assumptions. In contrast, my paper focuses on …rst-price CV
auctions. Incompleteness here arises from the range of possible rationalizing signal and value
distributions, instead of a ‡exible interpretation of bids. Hendricks, Pinkse and Porter (2003)
introduce nonparametric structural analyses to CV auctions. They use an ex post measure
of bidder values to test the assumption of equilibrium bidding. They also provide evidence
that the winner’s curse e¤ect dominated the competition e¤ect, leading to less aggressive
bidding in equilibrium as the number of bidders increase. Shneyerov (2006) introduces an
approach for counterfactual revenue analyses in common-value auctions without the need to
identify model primitives. In particular, he shows that for any given reserve price, equilibrium
bids from …rst-price auctions can be used to identify the expected revenues in second-price
auctions with the same reserve price. He also shows how to bound the expected gains in
revenues from English auctions under the general restriction of monotone value functions
and a¢ liated signals.
My paper makes three novel contributions. First, the focus on revenue distributions,
as opposed to distributional parameters such as expectations, allows more general revenue
analyses. Auction theory usually uses expected revenue as a criterion to compare auction
designs, but central tendency may not be justi…able in practice, say if the seller is not
risk-neutral. Knowledge of distributions is necessary for other criteria, such as maximizing
expected seller utility. (A seller may also choose a design to maximize the probability that
revenue falls in a certain range.) Second, bounds on revenue distributions can be constructed
for hypothetical reserve prices. One can then compare reserve prices within …rst-price or
second-price formats. In CV auctions, a counterfactual, binding reserve price r creates
serious challenges in policy analyses. The probability that no one bids higher than r in
equilibrium cannot be pinpointed from bids in the data, since the screening level can not
be identi…ed without further restrictions.8 Moreover, the mapping from equilibrium bids in
the data to those under the counterfactual r cannot be uniquely recovered. I address this
issue by bounding the bid that a marginal bidder under a counterfactual binding r actually
places in equilibrium under the data-generating auction format.9 These bounds in turn lead
to bounds on the revenue distribution under r. Finally, the bounds on revenue distributions
are robust and independent from exact forms of signal a¢ liations and value interdependence,
8
A screening level under r is the value of signal such that only bidders with signals higher than the
screening level will choose to submit bids above r in equilibrium. See Section 2 below for a formal de…nition.
9
A marginal bidder under r is the one whose signal is exactly equal to the screening level.
5
and are identi…ed from the distribution of equilibrium bids alone. This robustness comes
with the price of partial identi…cation of revenue distributions. Nonetheless, one can obtain
informative answers for some policy questions.
The remainder of the paper proceeds as follows. Section 2 introduces bounds on counterfactual revenue distributions in a benchmark model where data is collected from homogenous auctions with exogenous participation. Section 3 de…nes nonparametric estimators for
bounds and proves their pointwise consistency. Section 4 provides Monte Carlo evidence
about the performance of the bound estimators. Section 5 extends the benchmark model
to allow for observable auction heterogeneity and endogenous participation under binding
reserve prices in the data. Section 6 applies the proposed method to U.S. municipal bond
auctions on the primary market. Section 7 concludes.
2
Bounds on Counterfactual Revenue Distributions in
the Benchmark Model
This section focuses on a benchmark case where bids are observed from increasing, symmetric pure-strategy Bayesian Nash Equilibria (P SBN E) in homogenous single-unit …rst-price
auctions with a …xed number of bidders and no binding reserve prices. I show how to use
joint distributions of these bids to construct tight bounds on revenue distributions in counterfactual …rst-price and second-price auctions with a binding reserve price r > 0. Extensions
to cases where bids are observed from heterogenous auctions or auctions with endogenous
participation due to binding reserve prices are discussed in Section 5.
2.1
Model speci…cations
Consider a single-unit, …rst-price auction with N potential risk-neutral bidders and no reserve
price. Each bidder receives a private signal Xi but cannot observe his own valuation Vi . The
distribution of all bids submitted in equilibrium (denoted B0N
fBi0 gi=1;::;N ) is observed
from a random sample of independent, identical auctions, but neither Xi nor Vi can be
observed. For simplicity, Xi and Vi are both scalars. Throughout the paper I use upper case
letters to denote random variables, lower case letters for realized values, and bold letters for
vectors. The following assumptions are maintained for the rest of the paper.
6
A1 (Symmetric, A¢ liated Signals) Private signals XN fXi gi=1;::;N are a¢ liated with
N
support SX
[xL ; xU ]N , and the joint distribution FXN is exchangeable in all arguments.10
A2 (Interdependent Values) A bidder’s value satis…es Vi = N (Xi ; X i ), where N (:) is a
nonnegative, bounded, continuous function exchangeable in XN i fX1 ; :; Xi 1 ; Xi+1 ; :; XN g,
non-decreasing in all signals, and increasing in his own signal Xi over SX [xL ; xU ].
Note A2 implies private signals are drawn from identical marginal distributions on SX ,
and A1 includes private values (PV ) as a special case, where N (xi ; x i ) does not depend
N
on x i for all (xi ; x i ) 2 SX
. Common values (CV ) correspond to value functions that are
non-degenerate in rival signals X i . A pure strategy for a bidder under a given auction
structure (N; N ; FXN ) is a function b0i;N (:; N ; FXN ) : Xi ! R1+ . A pure-strategy Bayesian
Nash equilibrium is a portfolio fb0i;N (:)gi=1;::;N such that for all i, b0i;N (:) is the best response
to fb0j;N (:)gj2f1;::;N gnfig . (The superscript 0 signi…es that there is no reserve price.) That is,
for all i and x 2 SX ,
b0i;N (x) = arg max E(Vi bj
b
max
j2f1;:;N gnfig
b0j;N (Xj )
b; Xi = x) Pr(
max
j2f1;:;N gnfig
b0j;N (Xj )
bjXi = x)
The regularity conditions for existence of such a P SBN E is collected in A3 below. McAdams
(2006) proved A1,2,3 are su¢ cient for the existence of unique symmetric, increasing P SBN E
in …rst-price auctions. The restrictions in A3 are otherwise inessential for the main result
of partial identi…cation in this paper.
A3 (Regularity Conditions) (i) N (:) is twice continuously di¤erentiable; (ii) The joint
N
density of fXi gi=1;::;N exists on SX
, is continuously di¤erentiable, and 9flow ; fhigh > 0 such
N
that f (x) 2 [flow ; fhigh ] 8x 2 SX
.
De…nition 1 A joint distribution of bids fb0i;N gi=f1;::;N g in …rst-price auctions with no reserve price (denoted GB0N ) is rationalized by an auction structure f N ; FXN g if GB0N is
the distribution of bids in a symmetric, increasing PSBNE in the auction. Two structures
f N ; FXN g and f~N ; F~XN g are observationally equivalent if they generate the same distribution GB0N in symmetric, increasing PSBNE of …rst-price auctions. The identi…ed set
(relative to GB0N )is the set of all structures that are observationally equivalent given the bid
distribution GB0 .
10
Let Z be a random vector in RK with joint density f . Let _ and ^ denote respectively component-wise
maximum and minimum of any two vectors in RK . Variables in Z are a¢ liated if, for all z and z 0 in RK ,
f (z _ z 0 )f (z ^ z 0 ) f (z)f (z 0 ). For a more formal de…nition, see Milgrom and Weber (1982).
7
The …rst-order condition in PSBNE is characterized by:
b00
N (x) = [vh;N (x; x)
b0N (x)]
fYN jX (xjx)
FYN jX (xjx)
(1)
for all x 2 SX , where Yi;N
maxj2f1;:;N gnfig Xj , FYN jX (tjx)
Pr(Yi;N
tjXi = x), and
fYN jX (tjx) denotes the corresponding conditional density. And vh;N (x; y; N ; FXN ) is a bidder’s expected value conditional on winning with a pivotal bid, i.e. E(Vi jXi = x; Yi;N = y).
The equilibrium boundary condition is b0N (xL ) = vh;N (xL ; xL ). For notational ease, subscripts for bidder indices are dropped due to the symmetry in FXN and N . In an increasing
PSBNE where b00
N (:) > 0 on SX , Guerre, Perrigne and Vuong (2000) established a link
between the auction structure and GB0N by reformulating (1) using change-of-variable :
vh;N (x; x) =
b0N (x)
+
GMN0 jBN0 (b0N (x)jb0N (x))
gMN0 jBN0 (b0N (x)jb0N (x))
(b0N (x); GB0N )
(2)
maxj6=i b0N (Xj ) is the highest
where Bi;N
b0N (Xi ) is bidder i’s bid in equilibrium, Mi;N
tjb0N = b), and gMN0 jBN0 (tjb) is the correrival bid for bidder i, GMN0 jBN0 (tjb) = Pr(MN0
sponding conditional density.11 Again, indices for bidders are dropped due to symmetry.
Furthermore, subscripts N will also be dropped for the rest of this section and the following
section, as I focus on bid distributions from auctions with a …xed number of bidders.
2.2
Review of literature on P V auctions
In this subsection, I review the literature on identi…cation of signal distributions and optimal reserve prices in private value auctions. The objective is to highlight how unique
identi…cation of bidders’signal distributions and screening levels leads to exact knowledge
of the optimal reserve price. This motivates my approach of partial identi…cation when
the screening level can not be exactly pinned down in more general interdependent value
auctions.
Guerre, Perrigne and Vuong (2000) and Li, Perrigne and Vuong (2002) showed the joint
distribution of bidder values are nonparametrically identi…ed from distribution of equilibrium
bids in …rst-price, P V auctions with no reserve prices. This result holds regardless of the form
of dependence between private signals. The main idea is that in P V auctions, the left-hand
side of (2) is independent from the second argument (the highest rival signal) and therefore
11
Following a convention in the literature, I assume the second order conditions are always satis…ed and
thus …rst-order conditions are su¢ cient for characterizing the equilibrium.
8
can be normalized to the signal x itself. Hence the inverse bidding function can be fully
recovered from GB0 , for both independent and a¢ liated signals.12 Another simpli…cation
peculiar to P V auctions is that the screening level under a binding reserve price r is equal to
r itself. That is, bidders choose not to bid above r in equilibrium if and only if their private
signals are below r. To see this, note the screening level under r in a general interdependent
value auction is de…ned as :
x (r)
inffx 2 SX : E(Vi jXi = x; Yi
x)
rg
If E(Vi jXi = x; Yi
x) < r for all x 2 SX , let x (r) = xU . In P V auctions, E(Vi jXi =
x; Yi x) = E(Vi jXi = x) and the normalization E(Vi jXi = x) = x implies x (r) = r. Thus
in P V auctions, both the signal distribution FX and x (r) are exactly recovered from GB0 .
In principle, knowledge of FX in P V auctions is su¢ cient for …nding counterfactual
revenue distributions under a binding reserve price r. It follows that the optimal r which
maximizes the expected revenue is also identi…ed. Yet in reality it can be impractical to
implement this fully nonparametric estimation due to data de…ciencies, especially when the
signals are a¢ liated. Li, Perrigne and Vuong (2003) proposed a nonparametric algorithm for
estimating optimal reserve prices that is implemented with less intensive computations. The
idea is to express expected seller revenue under r as a functional of the observed distribution
of equilibrium bids and r. Then optimizing a sample analog of this objective function
over reserve prices gives a consistent estimator of the optimal reservation price. Again the
assumption of private values is indispensable for two reasons. First, it implies x (r) = r
under appropriate normalizations, which is used for de…ning the objective function; Second,
it ensures full nonparametric identi…cation of the distributions of counterfactual equilibrium
bidding strategies.
This approach cannot be applied to CV auctions with a¢ liated signals immediately because of two non-identi…cation results. First, the screening level cannot be pinned down
without further restrictions on how bidders’signals and valuations are correlated. Second,
inverse bidding functions can not be recovered without knowledge of . Hence underlying structure f ; FX g can not be identi…ed. These pose a major challenge for identifying
counterfactual revenue distributions in CV auctions.
12
In private value auctions, the conventional normalization of the signals is E(Vi jXi = x) = x for all x.
9
2.3
Observational equivalence of P V and CV
In this subsection I prove the observational equivalence of P V and CV paradigms when GB0
is observed from …rst-price auctions with a …xed number of bidders. That is, GB0 can be
rationalized by a P V structure if and only if it can be rationalized by a CV structure.13 This
preliminary question sheds lights on the scope of bid distributions where a uni…ed approach of
counterfactual analyses for both P V and CV structures is needed. To understand this point,
~ B0 that would be rationalized only by a P V structure
suppose there could exist certain G
but not any CV ones. In this case, the issue of …rst-order importance would be to derive
~ B0 , and then fully recover the underlying P V structures
testable implications of all such G
for those bid distributions satisfying such implications.
~ B0 cannot exist. Thus a robust approach
The rest of the subsection proves such a G
of counterfactual analyses that does not count on distinguishing P V and CV structures is
needed for any observed distributions of bids from a given auction structure. Let F denote
the set of joint signal distributions that satisfy A1, and
the set of value functions that
satisfy A2. Let CV denote a subset of that is non-degenerate in rival signals X i . Below
I give necessary and su¢ cient conditions for GB0 to be rationalized by some element of
F.
CV
Proposition 1 A joint distribution of bids GB0 observed in …rst-price auctions with nonbinding reserve prices can be rationalized by some f ; FX g 2 CV F if and only if (i) GB0 is
a¢ liated and exchangeable in all arguments; and (ii) (b; GB0 ) = b + GMi0 jBi0 (bjb)=gMi0 jBi0 (bjb)
is strictly increasing on the support of individual bids [b0L ; b0U ].
Li, Perrigne and Vuong (2002) showed conditions (i) and (ii) are also necessary and suf…cient for GB0 to be rationalized by some P V structure. It follows that a GB0 is rationalized
by some P V structures if and only if it is also rationalized by some 2 CV . This suggests
that researchers cannot distinguish P V and CV paradigm only using bid distributions from
homogenous auctions with a …xed number of bidders and no reserve prices.14
13
La¤ont and Vuong (1996) proved the su¢ ciency as they showed the non-identi…cation of CV auctions.
It remains unanswered whether the converse is true.
14
Recent literature in empirical auctions have developed ways to distinguish two structures with augmented data containing bids from more than one auction formats. These include the use of exogenous
variations in the number of bidders as in Haile, Hong and Shum (2003), ex post measures of bidder values
as in Hendricks, Pinkse and Porter (2003), and bid distributions under a strictly binding reserve price as in
Hendricks and Porter (2007).
10
2.4
Bounding revenue distributions in counterfactual 1st-price auctions
A conventional criterion for choosing optimal reserve prices is the expected revenue for the
seller. The Revenue Equivalence Theorem states that in auctions with independent private
values, optimal reserve prices are the same for both 2nd-price and 1st-price auctions, and are
independent from the number of potential bidders. On the other hand, there is no theoretical
result about the choice of optimal reserve prices in general 1st-price auctions with a¢ liated
signals, interdependent values, and a …nite number of bidders. The answer depends on the
speci…cs of model primitives and is open for empirical analyses. Besides, expected revenue
is not an appropriate criterion to use if the seller is not risk neutral. Knowledge of revenue
distributions in counterfactual auction formats helps address both concerns. For a binding
reserve price r, I propose informative bounds on FRI (r) that are constructed from GB0 alone.
2.4.1
The link between GB0 and FRI (r)
I start by establishing links between observed bid distributions GB0 and bidders’equilibrium
bidding strategies as well as the distribution of revenues in counterfactual 1st-price auctions
with a binding reserve price r. The equilibrium strategy in …rst-price auctions under a reserve
price r 0 has a closed form:
Z x
r
b (x; ; FX ) = rL(x (r)jx; FX ) +
vh (s; s; ; FX )dL(sjx; FX ) 8x x (r)
x (r)
br (x; ; FX ) < r 8x < x (r)
Rx
where L(sjx; FX ) expf s (u; FX )dug and (x; FX ) fY jX (xjx)=FY jX (xjx). This section focuses on a benchmark model where the bid distribution is observed from auctions
with a …xed number of bidders. Hence the superscript N is suppressed for notational ease.
For any given x on the closed interval SX , L(sjx; FX ) is a well-de…ned distribution function with support [xL ; x] and is …rst-order stochastically dominated by the distribution of
the second highest signal (i.e. FY jX (sjx)=FY jX (xjx)).15 The two distributions are identical
when bidders’private signals are i.i.d.. The range of r for nontrivial counterfactual analyses
is SRP
[ 0L ; 0U ], where 0k
(b0 (xk ); GB0 ) for k = L; U . This is because for r < 0L ,
x (r) = xL and there is no e¤ective screening of bidders, while for r > 0U , all bidders are
15
That L(:jx) is a well-de…ned distribution on [xL ; x] is shown in Krishna (2002). Furthermore L(sjx)
R x f (ujx)
F jX (sjx)
expf s FYY jX
dug = FYY jX
(xjx) , where the inequality follows from the fact that FY jX (xjz)=fY jX (xjz) is
jX (ujx)
decreasing in z for all x when signals are a¢ liated.
11
screened out with probability 1. Let v0 denote the seller’s own reserve value of the auctioned
asset. For all r 2 SRP and r > v0 , the distribution of revenue in counterfactual …rst-price
auctions with a binding reserve price r (denoted RI (r)) is:
FRI (r) (t; ) = 0 8t < v0
= PrfX (1) < x (r; )g 8t 2 [v0 ; r)
= PrfX (1)
r
(t; )g 8t 2 [r; +1)
where X (k) denotes the k-th highest out of N signals, r (t) denotes the inverse function of
the equilibrium strategy br (:) at a given bid level t, and 2
F denotes the underlying
structure (where
F is the set of primitives that satisfy A1 and A2 ).16 This distribution
would be exactly identi…ed from GB0 if a mapping between the strategy under r and that
with no reserve prices can be fully recovered from GB0 . For any rationalizable distribution
GB0 that satis…es conditions (i) and (ii) in Proposition 1, let (GB0 ) denote a subset of
structures in
F that rationalizes GB0 . For notational ease, the dependence of b0 and br
on the structure 2
F is suppressed.
Lemma 1 Consider a rationalizable GB0 . For all 2 (GB0 ) in …rst-price auctions with
any r 2 SRP and x x (r; ), br (x; ) = r (b0 (x; ); GB0 ) where r solves the di¤erential
equation
0
(b; GB0 ) = [ (b; GB0 )
(b; GB0 )] ~ (b; GB0 )
(3)
for b
b0 (x (r; ); ), with ~ (u; GB0 ) de…ned as
(b0 (x (r; ); ); GB0 ) = r.
gM 0 jB 0 (uju)
GM 0 jB 0 (uju)
and the boundary condition
Thus r can be constructed from the observed bid distribution GB0 up to an unknown
bid that a marginal bidder under r would place in an auction with no reserve prices (denoted
b0 (x (r))). This is not surprising, as binding reserve prices a¤ect bidding strategies in equilibrium only through the boundary condition br (x (r)) = r.17 My construction of bounds on
16
By de…nition Pr(RI (r) t) = 0 for all t < v0 . For all t 2 [v0 ; r), Pr(RI (r) t) = Pr(RI (r) = v0 ) =
Pr(X (1) < x (r)). Note br0 (x) > 0 for all r 2 SRP and x > x (r), and br (x (r)) = r. Hence br (x) is
invertible on [r; +1), and for t
r, PrfR(r)
tg = PrfX (1) < x (r)g + PrfX (1) 2 [x (r); (br ) 1 (t))g
(1)
r
= Pr(X
(t)) for all t 2 [r; +1).
17
To see this, note br and b0 are solutions to the di¤erential equation:
b0 (x) = [vh (x; x; ; FX )
b(x)]
fYN jX (xjx)
FYN jX (xjx)
with di¤erent boundary conditions b(x (r)) = r and b(xL ) = vh (xL ; xL ) respectively.
12
FRI (r) follows three intuitive steps. First, derive tight bounds on b0 (x (r)) from GB0 under
the general restriction of a¢ liated values and signals, and then substitute the bounds for
the unknown marginal bid in the boundary condition of (3) to get envelops on r . Next, for
any given level of counterfactual revenue t, invert these envelops to get a possible range of
b0 ( r (t)) (the hypothetical bid that a bidder who bids t under reserve price r would place
when there is no reserve price). Finally, use the distribution of winning bids in auctions with
no reserve prices to construct bounds on FRI (r) (t).
2.4.2
Bounds on counterfactual screening levels
I start by deriving the range of possible screening levels under a given binding reserve
price r. Let v(x; y)
E(Vi jXi = x; Yi
y), and vl (x; y)
E(vh (Y; Y )jXi = x; Yi
Ry
fY jX (sjx)
y) = xL vh (s; s) FY jX (yjx) ds. In symmetric Bayesian Nash equilibria, v(x; x) is the winner’s expected value if his signal is x (which is the same in both 1st-price and 2nd-price
auctions), and vl (x; x) is the winner’s expected payment in 2nd-price auctions with no reserve prices. For all x 2 SX and structures
2
F, a¢ liation between signals and
values implies vh (x; x; )
v(x; x; ), and the equilibrium condition in 2nd-price auctions guarantees v(x; x; )
vl (x; x; ). Let xl (r; )
arg minx2SX [vh (x; x; ) r]2 and
xh (r; ) arg minx2SX [vl (x; x; ) r]2 .
Lemma 2 (i) For all 2
F, x (r; ) 2 [xl (r; ); xh (r; )] for all r 2 SRP ; (ii) For any
rationalizable bid distribution GB0 , 9 2 (GB0 ) such that xl (r; ) = x (r; ) for all r 2
SRP . Furthermore, for all " > 0, 9 2 (GB0 ) such that supr2SRP jxh (r; ) x (r; )j ".
The bounds on the screening level are robust as they are constructed in a general environment where no restriction is placed on the form of value interdependence and signal
a¢ liations. In other words, the screening level can never fall outside the bounds, provided
bidders’values and private signals are a¢ liated. The true screening level coincides with the
upper bound if the winner …nds his rivals’signals uninformative about his own value. This
includes P V auctions as special cases. On the other hand, it hits the lower bound if the
margin between a winner’s own signal and the highest competing signal reveals no additional
information about his own value. For better intuition behind the bounds, consider special
cases where a bidder’s value function is additively separable between his own signal and the
vector of rival signals. Then the lower and upper bounds on the screening level correspond
to extreme cases of weights (1 and 0 respectively) that a bidder assigns to his own signals
while calculating the expected value conditional on winning.
13
Lemma 2 illustrates how the indeterminacy of underlying model structures leads to
a possible range of screening levels. However, it does not bound on the marginal bid directly, as both fxk (r; )gk=l;h and the bidding strategy b0 (:; ) depend upon the unknown
structure . Lemma 3 below proposes a way to bound the marginal bid by relating winners’
expected payment in 2nd-price auctions, as well as the equilibrium strategies, to observable
bid distribution GB0 . Let SB 0 denote the support of equilibrium bids in 1st-price auctions
b0 (xk ) for k = L; U .) Let
with no reserve prices. (That is, SB 0
[b0L ; b0U ] where b0k
Rb
gM 0 jB 0 (sjb)
(s; GB0 ) G 0 0 (bjb) ds for b b0L . For r 2 SRP , de…ne
l (b; GB0 )
b0 (xL )
M jB
b0l;r (GB0 )
arg minb2SB0 [ (b; GB0 )
r]2
b0h;r (GB0 )
arg minb2SB0 [ l (b; GB0 )
r]2
Lemma 3 Consider any rationalizable bid distribution GB0 . Then (i) for all r 2 SRP
and all
2 (GB0 ), b0 (x (r; ); ) 2 [b0l;r (GB0 ); b0h;r (GB0 )]; (ii) for all r 2 SRP and
b 2 [b0l;r (GB0 ); b0h;r (GB0 )), 9 2 (GB0 ) such that b0 (x (r; ); ) = b.
To understand these results, notice that and l can relate observed bid distributions
in equilibria to functionals of the underlying structures vh (:; ) and vl (:; ) through the
…rst-order conditions. More importantly, by construction, they only depend upon model
primitives f ; FX g through the observable GB0 generated in equilibria. Therefore, these
links between vh (:; ), vl (:; ) and GB0 are invariant over the identi…ed set of structures
(i.e. for all 2 (GB0 )). Then inverting these functions at a given binding counterfactual
reserve price r gives bounds on the marginal bid. The bounds are tight and sharp in the
sense that each point between the bounds depicts a marginal bid corresponding to a certain
structure within the identi…ed set (i.e. the set of observationally equivalent structures). In
other words, this range of possible marginal bids have exhausted all information that can
be extracted from the symmetric and a¢ liated properties of the values and signals, for it is
impossible to reduce the distance between the bounds in the absence of additional restrictions
on and FX .
2.4.3
Bounding the mapping between strategies and FRI (r)
Recall r solves (3) with an unidenti…ed boundary condition r (b0 (x (r; ); ); GB0 ) =
r. By replacing the unknown marginal bid in the boundary conditions with its bounds
fb0k;r (GB0 )gk=l;h , we can derive two solutions f r;k (:; GB0 )gk=l;h that are envelops on the
14
mapping between strategies b0 and br (only for bidders above the screening level under r).
Again this is due to the fact that reserve prices r (and therefore the marginal bid) enters
bidders’strategies only through boundary conditions. For any revenue level t considered in
a counterfactual 1st-price auction with a binding reserve price r, tight bounds on the hypothetical bid b0 ( r (t; ) ; ) (that a bidder who would bid t under r actually bids in auctions
with no reserve prices) can be derived by inverting the envelops r;k at t.
Lemma 4 Consider a rationalizable distribution GB0 . For k 2 fl; hg and r 2 SRP , let
f r;k (:; GB0 )gk=l;h denote solutions to the di¤erential equation (3) with boundary conditions
0
b0k;r (GB0 ). De…ne
r;k (bk;r (GB0 ); GB0 ) = r for b
1
r;k (t; GB0 )
arg
min
b2[b0k;r (GB0 );b0U ]
[
r;k (b; GB0 )
t]2 :
Then: (i) for any
2 (GB0 ), r;h (b; GB0 )
b0h;r (GB0 ) and
r (b; ; GB0 ) for all b
b0 (x (r; ); ), and r;k (:; GB0 ) are increasing on
r;l (b; GB0 )
r (b; ; GB0 ) for all b
[b0k;r (GB0 ); b0U ] for k = l; h; (ii) for any revenue t > r, and all b 2 [ r;l1 (t; G0B ); r;h1 (t; G0B )),
9 2 (G0B ) such that b0 ( r (t; ); ) = b.
The Lemma shows how sharp bounds on the marginal bid lead to sharp bounds on
the hypothetical bid b0 ( r (t; ); ) for all revenue level above the counterfactual reserve
price. This in turn will deliver the point-wise sharpness of bounds on revenue distributions
in counterfactual auctions below. A nice property of f r;k gk=l;h is that r;l
r;h is non1
1
0
18
increasing in b for b
b (xh (r)). This implies the length of [ r;l (t; GB0 ); r;h (t; GB0 )] is
decreasing in the revenue level t provided both r;l and r;h increase at a moderate rate.
Proposition 2 Consider a rationalizable GB0 and any r 2 SRP with r > v0 . Then for
all
2 (GB0 ), FRl I (r) (GB0 ) F:S:D: FRI (r) ( ) F:S:D: FRuI (r) (GB0 ), where F:S:D: denotes
…rst-order stochastic dominance, and
FRl I (r) (t; GB0 ) = 0
8t < v0
= Pr(b0 (X (1) ; ) < b0l;r (GB0 )
= Pr(b0 (X (1) ; )
18
Proof: Note
0
0
r;l (b; GB )
0
0
r;h (b; GB )
= ~ (b; G0B )
"Z
1
r;l (t; GB0 ))
b0 (xh (r))
b0 (xl (r))
as (b; G0B )
r for all b
8t 2 [v0 ; r)
r
8t 2 [r; +1)
#
0
0
~
~
~
(b; GB )dL(bjb; GB )
0
b0 (xh (r)) in equilibrium. The inequality is strict if b0 (xh (r)) > b0 (xl (r)).
15
and
FRuI (r) (t; GB0 ) = 0
8t < v0
= Pr(b0 (X (1) ; ) < b0h;r (GB0 )
= Pr(b0 (X (1) ; )
8t 2 [v0 ; r)
1
r;h (t; GB0 ))
8t 2 [r; +1)
The distribution of the highest bid is observed from auctions with no reserve prices.
Therefore the bounds can be nonparametrically constructed from GB0 . Furthermore, it follows from Lemma 3 and Lemma 4 that for a given revenue level t, any point within the
interval [FRl I (r) (t; GB0 ); FRhI (r) (t; GB0 )) correspond to certain true distribution of counterfactual revenue FRI (r) (t; ) for some 2 (GB0 ). In other words, fFRk I (r) (t; GB0 )gk=l;h form
point-wise tight, sharp bounds on the true revenue distributions in counterfactual …rst-price
auctions with a binding reserve price r.
2.4.4
A simpler upper bound of FRI (r)
Below I propose a simpler upper bound on FRI (r) (denoted F~RuI (r) ) that can be constructed
using observed revenue distributions from auctions with no binding reserve prices (denoted
FRI (0) ) as opposed to from GB0 . De…ne
F~RuI (r) (t; FRI (0) ) = 0, 8t < v0
= Pr(b0 (X (1) ; ) < r), 8t 2 [v0 ; r)
= Pr(b0 (X (1) ; )
t), 8t
r
This simpler upper bound is easy to construct, and is depicted in Graph 1 in the appendix.
It coincides with FRuI (r) when private signals are independent and identically distributed.
However, if there is strict a¢ liation among the signals, the simpler upper bound will be
less e¢ cient than FRuI (r) in the sense that F~RuI (r) fails to rule out some of the counterfactual
revenue distributions that can not be rationalized by any element in the identi…ed set.
Proposition 3 Consider any rationalizable distribution GB0 . Then for all 2 (GB0 ) and
r 2 SRP , FRI (r) ( ) F:S:D: FRuI (r) (GB0 ) F:S:D: F~RuI (r) (FRI (0) ( )). Furthermore, FRuI (r) (GB0 ) =
F~RuI (r) (FRI (0) ( )) if private signals are independently, identically distributed.
The proof uses the fact that bidders with signals above the screening level x (r) would bid
higher under the counterfactual binding r than in auctions with no binding reserve prices.
16
Intuitively, it is not surprising that F~RuI (r) is in general less e¢ cient than FRuI (r) , since the
latter uses full information from GB0 while the former only uses GB0 indirectly through a
functional of FRI (0) . Nonetheless, this simpler upper bound is still interesting for two reasons.
First, the i.i.d. restrictions on private signals is testable in symmetric equilibria using the bid
distribution observed. Hence in practice when signals are tested to be i.i.d., the simpler upper
bound are known to be e¢ cient. Second, when signals are not i.i.d., comparing FRuI (r) (GB0 )
and F~RuI (r) illustrates how the a¢ liation of private signals helps narrow down the scope of
possible counterfactual revenue distributions corresponding to the identi…ed set.
2.5
Bounding revenue distributions in counterfactual 2nd-price
auctions
This subsection construct bounds on counterfactual revenue distributions in 2nd-price auctions under reserve price r (denoted FRII (r) ) from GB0 . Theory predicts for any given reserve
price r, the expected revenues in 2nd-price auctions are at least as high as those in 1st-price
auctions provided bidder signals are a¢ liated. However, the size of this di¤erence is an open
empirical question. In addition, within the format of 2nd-price auctions, theory is silent
about the choice of optimal reserve price r that maximizes expected revenue when signals
are a¢ liated. Knowledge of FRII (r) would help address these open questions.
The equilibrium strategy in 2nd-price auctions under a binding reserve price r is
r
r
(x; ) = vh (x; x; ) 8x
x (r; )
(x; ) < r 8x < x (r; )
Consider any structure 2
F. For all r 2 SRP and r > v0 , the distribution of revenues
in a second-price auction with reserve price r is:19 (for notational ease, dependence of vh and
x (r) on the structure is suppressed.)
FRII (r) (t; ) = 0 8t < v0
= Pr(X (1) < x (r)) 8t 2 [v0 ; r)
= Pr(X (2) < x (r)) 8t 2 [r; vh (x (r); x (r)))
= Pr(vh (X (2) ; X (2) )
t) 8t 2 [vh (x (r); x (r)); +1)
The link between GB0 and revenue distributions in counterfactual 2nd-price auctions is
easier to see than in 1st-price auctions, since the distribution of b0 (X (1) ; ) and b0 (X (2) ; )
19
See the proof of Proposition 3 below for details.
17
are both observed. Besides, the bids in 2nd-price auctions with a counterfactual reserve
price r can be exactly recovered for bidders that are known to be unscreened under r. This is
because in Bayesian Nash equilibrium, vh (X; X; ) = (b0 (X; ); GB0 ) for all 2 (GB0 ).
However, the non-identi…cation of the marginal bid b0 (x (r)), and therefore the expected
value for the pivotal winner vh (x (r); x (r)), makes it impossible for researchers to fully
recover FRII (r) . Fortunately, just as in the case of 1st-price auctions, replacing the marginal
bid with fb0k;r (GB0 )gk=l;h in the de…nition of FRII (r) leads to point-wise bounds on FRII (r) .
Proposition 4 Consider a rationalizable distribution GB0 , and any r 2 SRP with r > v0 .
Then FRl II (r) (t; GB0 ) F:S:D: FRII (r) (t; ) F:S:D: FRuII (r) (t; GB0 ) for all 2 (GB0 ), where
FRl II (r) (t; GB0 ) = 0
8t < v0
= Pr(b0 (X (1) ; ) < b0l;r (GB0 ))
= Pr( (b0 (X (2) ; ); GB0 )
t)
8t 2 [v0 ; r)
8t 2 [r; +1)
and
FRuII (r) (t; GB0 ) = 0
8t < v0
= Pr(b0 (X (1) ; ) < b0h;r (GB0 ))
= Pr(b0 (X (2) ; ) < b0h;r (GB0 ))
= Pr( (b0 (X (2) ; ); GB0 )
t)
8t 2 [v0 ; r)
8t 2 [r; (b0h;r (GB0 ); GB0 ))
8t 2 [ (b0h;r (GB0 ); GB0 ); +1)
The intuition of the proof is demonstrated in Graph 2. To understand this proposition,
note by the de…nition of the identi…ed set of structures, the highest and second-highest
order statistics of equilibrium bids must be invariant among all 2 (GB0 ). Hence the
bounds fFRk II (r) gk=l;u are functionals of the observed bid distribution GB0 only. In addition, just as with 1st-price auctions, it follows from the sharpness of fb0k;r (GB0 )gk=l;h that
fFRk I (r) (t; GB0 )gk=l;h form point-wise tight, sharp bounds on the revenue distributions in
counterfactual 2nd-price auctions with a binding reserve price r.
3
Nonparametric Estimation of Bounds
In this section, I de…ne three-step estimators fF^Rk I (r) gk=l;u for bounds on FRI (r) and FRII (r) .
The basic idea is to replace GB0 with its sample analog in estimation. I consider the case
18
where data reports all bids submitted in Ln independent, homogenous auctions, each with
n potential bidders and no reserve prices.20
Let SB
[b0L ; b0U ] denote the support of bids observed in 1st-price auctions with nonbinding reserve prices. For all (m; b) 2 SB2 , de…ne:
^ M;B (m; b) = 1 PLn 1 Pn 1(mil m)KG ( b bil )
G
Ln hG l=1 n i=1
hG
P
P
m mil b bil
1
Ln 1
n
;
)
g^M;B (m; b) =
l=1
i=1 Kg (
2
Ln hg
n
hg
hg
where bil and mil are bidder i’s bid and the highest competing bid against him in auction l,
Ln is the total number of auctions with n potential bidders, KG and Kg are symmetric kernel
functions with bounded hypercube supports with each side equal to 2, and hg , hG are the
corresponding bandwidths. It is well known that density estimators are asymptotically biased
near boundaries of the support for some b 2 [b0L ; b0L + hg ) [ (b0U hg ; b0U ]. Let
max(hg ; hG )
0
0
and SB; = [bL + ; bU
] be an expanding subset of SB (as sample size increases) where
^ M;B and g^M;B are asymptotically unbiased. A natural estimators for SB; is:
G
S^B;
[~bL ; ~bU ], where ~bL = ^bL + ; ~bU = ^bU
where ^bL = mini;l bil and ^bU = maxi;l bil converge almost surely to b0L and b0U respectively.
Nonparametric estimators for and l are de…ned as:
Z b
^
^(b) = b + GM;B (b; b) ; G
^ M;B (~bL ; b)
~ M;B (b; b) =
g^M;B (t; b)dt + G
g^M;B (b; b)
~bL
Z b
~
^
^(t) g^M;B (t; b) dt
^ (b) = ^(~bL ) GM;B (bL ; b) +
l
~ M;B (b; b)
^ M;B (b; b)
~bL
G
G
~ M;B and ^l are de…ned over the random support S^2 and S^B; respectively. The
where G
B;
…rst-step estimators for fb0k;r (GB0 )gk=l;h are de…ned as:
^b0 = arg min ^ [^(b)
l;r
b2S ;B
r]2 ;
^b0 = arg min ^ [^ (b)
h;r
b2S ;B l
r]2
In the second step, I …rst construct kernel estimator for r;l (b) and r;h (b) on S^ ;B using
…rst-step estimates ^b0l;r and ^b0h;r . For k = fl; hg, de…ne:
Z b
0
^r;k (b; ^b0 )
^
^(t) ^ (t)L(tjb)dt
^
^
rL(bk;r jb) +
8b 2 (^b0k;r ; ~bU ]
k;r
^b0
k;r
r
20
8b 2 [~bL ; ^b0k;r ]
"Independence" here has both economic and statistical interpretations. First, there is no strategic
interaction or learning across the auctions, so that the same …rst-order condition characterizes equilibria in
all auctions. Second, the random vectors of bidders’ private information are independent across auctions.
"Homogeneity" means the auctioned object in all auctions have the same commonly observed characteristics.
19
where ^ (t)
are:
^ M;B (t; t) and L(tjb)
^
g^M;B (t; t)=G
^ 1 (t) = arg min ^ [^r;l (b)
r;l
b2SB;
By de…nition, S^ ;B
as:
S
;B
t]2 ;
exp(
Rb
t
^ (s)ds). Estimators for
^ 1 (t) = arg min ^ [^r;h (b)
r;h
b2SB;
1
r;k (t; GB0 )
t]2
with probability 1. In the …nal step, bounds on FRI (r) are estimated
1 XLn
1(Blmax
F^Rl I (r) (t) =
l=1
Ln
^ 1 (t));
r;l
1 XLn
F^RuI (r) (t) =
1(Blmax
l=1
Ln
^ 1 (t))
r;h
where Blmax = maxi=1;::;n bil is the highest bid in auction l.
k
The three-step estimators above converge in probability to the true bounds FR(r)
(t; GB0 )
over all r and t > r. Below I strengthen restrictions in A1-3 to include all regularity
conditions needed for consistency.
n
S1 For n
2, (i) The n-dimensional vectors of private signals (x1l ; x2l ; ::xnl )Ll=1
are
independent, identical draws from the joint distribution F (x1 ; ::; xn ), which is exchangeable
in all n arguments and a¢ liated with support SX
[xL ; xU ]; (ii) F (x1 ; ::; xn ) has R + n,
n
R
2, continuous bounded partial derivatives on SX , with density f (x)
cf > 0 for all
n
x 2 SX .
n
! R+ is positive, bounded, and continuous on
S2 (i) The value function n (:) : SX
the support; (ii) n (:) is exchangeable in rival signals X i , non-decreasing in all signals, and
increasing in own signal Xi over SX . (iii) n (:) is at least R times continuously di¤erentiable
d
and (xL ) > 0; (iv) vh (xU ; xU ) < 1 and dX
vh (X; X)jX=xU < 1.
In addition to maintaining the identifying restrictions of a¢ liation and symmetry among
signals and values in A1-3, the stronger version S1-2 also includes additional regularity
conditions on the smoothness of model primitives f and . This will lead to smooth properties
of bid distributions in equilibrium, which in turn, determines asymptotic properties of nonparametric estimators.
S3 (i) The kernels KG (:) and Kg (:) are symmetric with bounded hypercube supports of
R
R
~ b)dBdb
~
sides equal to 2, and continuous bounded …rst derivatives; (ii) KG (b) = 1, and Kg (B;
= 1; (iii) KG (:) and Kg (:) are both of order R + n 2.
These are standard assumptions on kernels necessary for proving the asymptotic properties of kernel estimators.
Proposition 5 Let hG = cG (log L=L)1=(2R+2n
5)
and hg = cg (log L=L)1=(2R+2n
4)
, where cG
20
and cg are constants. Suppose S1-3 are satis…ed and R > 2n
p
t r, F^Rk I (r) (t) ! FRk I (r) (t) for k = l; u.
1, then for all r 2 SRP and
The proof is included in Appendix B, and proceeds in several steps. First, I prove
smoothness of bid distributions in equilibrium. Second, I show the kernel estimators ^l
and ^ converge in probability to l and uniformly over S^B; . Then I use a version of the
Basic Consistency Theorem in Newey and McFadden (1994) that is generalized for extreme
p
estimators with the objective functions being de…ned on random support) to show ^b0k;r ! b0k;r
p
for k = l; h and the relevant reserve prices. Next, I prove ^k;r (:; ^b0k;r ) ! k;r (:; b0k;r ) uniformly
1
p
over S^B; and again used the generalized Basic Consistency Theorem to show that ^ (t) !
r;k
1
r;k (t)
for all relevant t. Finally, I use the Glivenko-Cantelli uniform law of large numbers to
1
show empirical distributions of B max evaluated at ^ (t) for k = l; h are consistent estimators
r;k
l
for bounds on FRI (r) (t).
Estimating bounds on revenue distributions in counterfactual 2nd-price auctions follows
similar logic and is straightforward given the constructions above. De…ne:
1 XLn
(1:n)
F^Rl II (r) (t) =
1(Bl
< ^b0l;r )
8t 2 [v0 ; r)
l=1
Ln
1
1 XLn
(2:n)
=
1(Bl
< ^ (t))
8t 2 [r; +1)
l=1
Ln
and:
1 XLn
(1:n)
1(Bl
< ^b0h;r )
8t 2 [v0 ; r)
l=1
Ln
1 XLn
(2:n)
=
1(Bl
< ^b0h;r )
8t 2 [r; ^(^b0h;r ))
l=1
Ln
1
1 XLn
(2:n)
1(Bl
< ^ (t))
8t 2 [^(^b0h;r ); +1)
=
l=1
Ln
F^RuII (r) (t) =
1
where ^b0k;r is de…ned above and ^ (t) arg minb2S^B; [^(b) t]2 for t r. Pointwise consistency of F^Rk II (r) (t) for r
v(xL ; xL ) and t
r follows directly from similar arguments for
p
k
0
consistency of F^ I (t), and the fact that ^(^b ) ! (b0 ; GB0 ) = vh (xh (r); xh (r); ) for all
R (r)
2
4
h;r
h;r
(GB0 ).
Monte Carlo Experiments
Bounds on revenue distributions in counterfactual …rst-price and second-price auctions are
e¢ cient in that they have exhausted all information that can be extracted from GB0 . It is
21
impossible to derive a tighter range of possible counterfactual revenue distributions without
introducing further restrictions on how values and signals are related. Exactly how informative these bounds can be is determined by the unidenti…ed underlying primitives. In
this section, I report analytical as well as Monte Carlo evidence on the performance of our
bound estimators in …nite samples. The objective is to illustrate how the widths of estimated
bounds vary with structural parameters such as the a¢ liation between private signals, the
number of potential bidders n and the reserve price r.
4.1
Analytical impacts of signal a¢ liations
I start with an example where the impact of signal a¢ liations on how bounds on the allscreening probabilities (the probability that at no bidder bids above the reserve price in
counterfactual formats) can be studied analytically. I use a parametric design where signal
a¢ liations can be controlled.
Design 1 ( n = 2 with pure common values and a¢ liated signals) Two potential bidders
compete in an auction with Vi = (X1 +X2 )=2 for i = 1; 2. Private signals are noisy estimates
of a common random variable, i.e. Xi = X0 + "i for i = 1; 2. For either bidder, his noise
"i is independent from (X0 ; " i ), and distributed uniformly on [ c; c] for some 0 c 0:5.
The common random term X0 is distributed uniformly on [c; 1 c].
The signals have well-de…ned marginal densities with a simple form on [0; 1]. For example,
for c = 0:25, the density function is f (x) = 4x for 0 x 0:5 and 4 4x for 0:5 x 1.
(See Simon (2000) for details.) And their correlation coe¢ cient is:
corr(X1 ; X2 ) =
var(X0 )
(1 2c)2
=
var(X0 ) + var("i )
(1 2c)2 + 4c2
By de…nition, vh (x; x) = x, vl (x; x) = E[X2 jX2
x; X1 = x], and v(x; x) = x+vl2(x;x) . In
this design, vl (x; x) has a closed form, and the impacts of signal correlations on the widths
of bounds on the all-screening probability can be studied analytically. The derivation of the
closed form of vl (x; x; c) is included in the appendix.
Figure 1(a) plots vl (x; x; c) and vh (x; x; c) for c = [0:1 0:2 0:3 0:4]. The distance between
vh and vl is non-decreasing in private signals, as vl (x; x) is a truncated expectation and cannot
increase faster than the threshold x itself. Figure 1(b) plots the boundwidth xh (r; c) xl (r; c)
as a function of reserve prices for each c. For any given reserve price, bounds on screening
levels are narrower as c decreases and correlation increases. Intuitively, this is due to the
22
fact that conditional on winning with a pivotal bid, the di¤erence between the winner’s
expected value and his expected payment in a 2nd-price auction decreases as the signals are
increasingly positively correlated. When c = 0:1 and c = 0:2, the boundwidths are invariant
for r high enough. For di¤erent signal correlations, Figure 1(c) plots the size of bounds on
all-screening probabilities in 1st-price auctions as functions of counterfactual reserve prices
considered. That is, FX (1:2) (xh (r; c); c) FX (1:2) (xl (r; c); c), where X (1:2) is the higher of two
private signals. As the reserve price increases, the size of the bounds are unambiguously
smaller when signals have higher positive correlations. This is explained by the pattern in
Figure 1(b) and the distribution of X (1:2) as plotted in Figure 1(e). Note the probability
mass of X (1:2) is more skewed to the left when signals are less positively correlated. For a
low reserve price r, both xl (r; c) and xh (r; c) are small and the bounds on the screening level
under r are very close in size for all c. On the other hand, as Figure 1(e) shows, X (1:2) has
more probability mass close to 0 for higher positive signal correlations. Hence for r that are
low enough, the size of bounds on the all-screening probabilities in 1st-price auctions is bigger
for c = 0:1. For higher reserve prices r considered, the widths of bounds on the screening
level is greater for higher c (and smaller positive correlations). Besides, the probability mass
of X (1:2) is greater in the relevant range as signals are less positively correlated. Therefore,
the size of bounds on all-screening probabilities in 1st-price auctions under a higher r are
much bigger for auctions with less correlated private signals.
Figure 1(d) plots the widths of bounds on all-screening probabilities in counterfactual
2nd-price auctions as functions of r considered. That is, FX (2:2) (xh (r; c); c) FX (2:2) (xl (r; c); c)
for various c 2 [0:1; 0:5]. In this case, the boundwidths associated with a smaller c is almost
unambiguously smaller than those with higher c (and smaller correlations). Likewise, the
pattern is explained by arguments as demonstrated in Figure 1(b) and the distribution of
X (2:2) plotted in Figure 1(f). An obvious departure from the case of 1st-price auctions is
that, for auctions with less correlated private signals, the widths of bounds on all-screening
probabilities increase faster as r increases, reaches their peaks around r 2 [0:3; 0:4] and
then decreases faster than in the case of 1st-price auctions. As Figure 1(f) suggests, this is
explained by the fact that when c is higher, there is more probability mass of X (2:2) around
the center of the support, and the tail of the distribution diminishes faster.
4.2
Performance of F^Rk I (r) under i.i.d. signals
This subsection focuses on the performance of three-step estimators F^Rk I (r) when private signals are identically and independently distributed. The i.i.d. restriction can be tested in
23
empirical analysis by checking the identity of marginal bid distributions and the independence of their joint distribution. Also the i.i.d. assumption helps simplify the estimation
procedures. In this subsection, I vary n, r and distributional parameters and study their
impacts on estimator performances.
Design 2 ( n
3 with PCV and i.i.d. uniform signals) Private signals fXgi=1;::;n
are identically, independently distributed as uniform on [0; 1]. The pure common value is
P
Vi = nj=1 Xj =n.
Design 3 ( n 3 with PCV and i.i.d. truncated normal signals) Private signals fXi gi=1;::;n
are identically, independently distributed as truncated normal on [0; 1] with underlying paraP
meters ( ; 2 ). The pure common value is Vi = nj=1 Xj =n.21
The two designs …t in the general framework of symmetric, interdependent value auctions,
as independence is a special case of a¢ liation. Note the distribution of the average of signals
depends on n, and therefore the number of bidders are not exogenous to bidders values.
Hence both designs do not meet necessary restrictions for tests distinguishing P V and CV
auctions in Haile, Hong and Shum (2003). Thus it is appealing to adopt our robust approach
of partial identi…cation for these two designs, which does not require distinction between the
two paradigms. I experiment with di¤erent numbers of potential bidders and reserve prices
for Design 2. For each (n; r), I calculate the nonparametric estimates of F^Rk I (r) from 1; 000
simulated samples, each containing equilibrium bids submitted in 500 …rst-price auctions.
For Design 2,
n 1 1 1
b0n (x) =
( + )x
n n 2
and bids are simulated as random draws from a uniform distribution on [0; nn 1 ( n1 + 21 )].
For Design 3, I vary distributional parameters and in addition to n and r. For each
(n; r; ; ), I replicate the estimator for 1; 000 times, each based on a simulated sample of
500 auctions. For Design 3,
Z x
2
n 2
FXn 1 (s)
0
bn (x) =
s+
'(s)d n 1
n
FX (x)
xL n
where '(x) =
(x
(
x
xL
)
(
)
x
( L
)
)
and
FX (s)
FX (x)
=
(s
)
(
xL
)
x
)
(
xL
)
(
. Equilibrium bids are sim-
ulated by …rst drawing 500 n signals xil randomly from the truncated distribution, and
calculating b0;n (xil ) through numerical integrations. I use the classical approach of midpoint
approximations for numerical integrations for the rest of the paper. For both designs and
21
This form of value functions introduces a restriction (normalization) on signals, as it requires support
of signals to be the same as that of values.
24
each r, the true counterfactual revenue distribution FRI (r) can be recovered by inverting br (:),
which can be calculated with knowledge of their closed forms above. In symmetric equilibria,
bids under both designs are i.i.d.. This can be tested using the distribution of bids observed,
G0 (b)
and in practice simpli…es our estimation as (b; G0Bn ) = b + n 1 1 g0Bn(b) and l (b; G0Bn ) = b.
Bn
^0
P n 1 Pn
1 GBn (b)
^
^
The simpli…ed estimator is (b) b + n 1 g^0 (b) , where GBn (b) = L1n Ll=1
b),
i=1 1(bil
n
Bn
P
P
n
bil b
n 1
g^n0 (b) = Ln1hg Ll=1
i=1 K( hg ) and Ln is the number of auctions with n bidders. For
n
35
(1 u2 )1(juj
1).22 Bandwidths hg is
estimation, I use the tri-weight kernel K(u) = 32
1
2:98 1:06^ b (nLn ) 4n 4 , where ^ b is the empirical standard deviation of bids in the data.
The bandwidths are chosen in line with the consistency proposition in the appendix, while
the constant factor 1:06^ b is chosen by the "rule of thumb". (See Li, Perrigne and Vuong
(2002) for an example.) The multiplicative factor 2:98 is due to the use of tri-weight kernels.
(See Hardle (1991) for details.)
Figure 2 plots the true revenue distribution FRI (r) in Design 2 and, for di¤erent n and
r, reports the 5th percentile of F^Rl I (r) and the 95th percentile of F^RuI (r) out of 1; 000 pairs of
estimates. The two percentiles form an estimate of a conservative 90% pointwise con…dence
interval for the bounds [FRl I (r) ; FRuI (r) ]. (See Haile and Tamer (2003) for an example.) The
true revenue distribution always falls within the interval. The intervals for a lower r are narrower, holding n constant. On the other hand, more potential bidders correspond to tighter
con…dence regions ceteris paribus. To understand the pattern, note the boundwidth of the
n 1
2n
all-screening probability is Pr(b0n (X (1:n) ) r) Pr(b0n (X (1:n) )
r) = FX (1:n) ( nn 1 n+2
r)
n
2n
2n
FX (1:n) ( n+2
r), which is increasing in r for a given n. For a given r, n 1 1 n+2
r decreases in n
2n
and this o¤sets the impacts of a rising n+2 r and a more left-skewed FX (1:n) as competition
increases. The simulations suggest variations in the width of estimated con…dence intervals
are mostly due to impacts of n and r on the boundwidths of FRI (r) .
Figure 3 reports FRI (r) and estimates of conservative 90% con…dence intervals for Design
3. Again, the true revenue distribution falls within 90% point-wise conservative con…dence
intervals for the parameters considered. The impacts of n and r on the estimated con…dence
intervals in Design 3 are the same as those for Design 2 in Figure 2. In addition, Figure
3 also shows impacts of distributional parameters and on con…dence intervals. First,
holding n, r and …xed, the con…dence intervals become narrower as increases. This is
because for all t, E(XjX t) gets closer to t as the distribution of X is more skewed to the
left. Consequently, x (r) decreases for a given r, while the distance between vh and vl also
22
0
The triweight kernel is of order 2. In principle when n
3, kernels used in g^Bn
should be of higher
order. But can lead to the issue of negative density estimates. Therefore empirical literature typically ignore
this requirement and use kernels with order 2.
25
becomes smaller. As a result, the bound on the all-screening probability is shifted to the left
and becomes tighter. Second, the impact of on con…dence intervals depends on , holding
n and r …xed. A higher standard deviation increases the width of con…dence intervals for
signal distributions su¢ ciently skewed to the left, but reduce the width of con…dence intervals
for signal distributions su¢ ciently skewed to the right. The impacts are more obvious for
distributions skewed to the right. This pattern is explained by similar reasons above. Again,
simulations suggest variations in the width of estimated con…dence intervals are mostly due
to impacts of n and r on the size of bounds on FRI (r) .
4.3
Performance of F^Rk I (r) with a¢ liated signals
When signals are not independently and identically distributed, there are no simpli…ed forms
for ^ and ^l , and the full nonparametric estimates in Section 3 apply. In this subsection I
P
extended Design 1 for n 3 so that Vi = nj=1 Xj =n, and experiment with the correlation
parameter c to study its impact on the performance of estimators. With n
3, it is
impractical to derive the analytical form of the inverse hazard rate fY jX;n (uju)=FY jX;n (uju).
To …nd out the true revenue distribution, I replace vh (x; x; c) and L(sjx; c) with their kernel
estimates in a simulated sample of 5 105 auctions, and calculate the equilibrium bidding
strategies using these estimates and numerical integrations. The true counterfactual revenue
distribution FRI (r) is then recovered with knowledge of the distribution of the highest signal
X (1:n) . For each (c; n), I simulate 200 samples, with each containing 1; 000 simulated …rstprice auctions. For each r and revenue level t, Figure 4 reports the point-wise 5-th percentile
of F^Rl I (r) (t) and the 95-th percentile for F^RuI (r) (t) out of 200 pairs of estimates. This forms
estimates for a conservative 90% con…dence interval for the bounds on FRI (r) . Figure 4
shows the true FRI (r) lies within the estimated con…dence interval for r = 0:2 or 0:5, c = 0:2
or 0:4 and n = 3 or 4. Holding r and c constant, the widths of the estimated con…dence
intervals decrease slightly as n increases. For r = 0:2, higher correlation leads to slightly
wider con…dence intervals, whereas for r = 0:5 higher signal correlation leads to obviously
narrower con…dence intervals. Smaller correlations among signals implies the distribution
of X (1:n) is more skewed to the left, and the distance between vl and vh are bigger. These
explain why a higher c leads to wider con…dence intervals when r is high at 0:5. On the
other hand, when r is low at 0:2, the left-skewness of FX (1:n) o¤sets the impact of a wider
bound [xl (r; c); xh (r; c)] due to a higher c, and may lead to a narrower con…dence interval.
Furthermore, the theory also states for x
x (r; c) the bounds on r (b0 (x; c)) is tighter
as b0 (x; c) increases. For t > r, this counteracts the left skewness of FX (1:n) due to lower
26
correlations. This is consistent with patterns in Figure 4 where con…dence intervals on
FRI (r) never broaden substantially as revenue level t increases.
5
5.1
Extensions
Heterogenous auctions
In practice, bids are often collected from heterogenous auctions with varied characteristics
of the objects for sale. If commonly observed by all bidders, such heterogeneity a¤ects
bidders’strategies and revenue distributions in counterfactual auctions. If researchers can
completely control for heterogeneity across auctions by using the observables in the data,
then logic for bounds on revenue distributions in homogenous counterfactual auctions extends
immediately. Speci…cally, auctions are homogenous within subsets of the data if such features
(denoted Z) are controlled for, and the same algorithm in the benchmark model extends to
bounds on revenue distributions conditional on such characteristics FRI (r)jZ=z , which can be
constructed from conditional bid distribution G0BjZ=z . However, real challenges can arise
from observable auction heterogeneity is empirical implementation. Constructing bounds on
conditional revenue distributions requires a large cross-sectional data of homogenous auctions
with …xed features z and a …xed number of potential bidders n. This issue of data de…ciency
aggravates as the dimension of z increases. Below I show if signals are independent from
observable heterogeneities conditional on n, and are additively separable from the latter in
the value functions, then it is possible to "homogenize" bids across heterogenous auctions,
thus alleviating the data de…ciency problem.
A1’ (Interdependent Values) Vi;N = h(Z0 ) + N (Xi ; X i ), where h(:) is di¤erentiable,
and N is bounded, continuous, exchangeable in its last N 1 arguments, non-decreasing in
all arguments, and increasing in Xi .
A4 (Conditional Independence of X and Z) Conditional on N = n, fXi gi=1;::;n is independent from Z.
Then a PSBNE in the auction with no binding reserve price is a pro…le of strategies that
solve:
b0i (x; z; n) = arg max E[(Vi
b
b)1fmax b0j (Xj ; Z)
j6=i
bgjXi = x; Z = z; N = n]:
27
Under these restrictions, common knowledge of auction features impact strategies of all
bidders in the same way. As the proposition below shows, the separability and the index
speci…cation of value functions are inherited by bidding strategies in equilibria.
Proposition 6 Under A1’, A2, A3 and A4, bidders’equilibrium strategies satisfy : b0i (x; z; n)
Rx
= h(z0 ) + (x; n) for all x; z and i, where (x; n)
(s; n)dL(sjx; n), and (s; n)
xL
E[ (X)jXi = Yi = s; N = n].
Fix the number of potential bidders n, the proposition implies E(b0i jZ = z; N = n) =
h(z ) + E( (X; N )j N = n), where the second term is a constant independent from Z.
This becomes a single index model, and both Powell, Stock and Stocker (1989) and Ichimura
(1991) showed can be
identi…ed up to scale, and estimated consistently using average
derivative estimators or semiparametric least square estimators. In the special case where
h(:) is known to be the identity function, an OLS regression of bids from heterogenous
auctions on the characteristics z for a …xed n will estimate consistently. Alternatively,
including dummies for the number of potential bidders in a pooled regression will also give
consistent coe¢ cient estimators for .
0
A corollary of the proposition is that for any pair of di¤erent features of auctions z and z,
the equilibrium strategies for a given signal x are related as b0 (x; z; n) = b0 (x; z; n) h(z0 )+
h(z0 ). Thus when h is known, bids across heterogenous auctions can be "homogenized" at
any speci…c reference level z so that more observations are available for estimating G0B(Z) .
Larger sample size leads to better performance of estimators of bounds on FRI (r)jZ=z .
5.2
Binding reserve prices in data
In practice, bids are often collected from homogenous auctions under a commonly known
reserve price r that is high enough to have a positive probability of screening out some of
the bidders. This gives rise to new challenges relative to benchmark cases where there is no
binding reserve price in the data. First, bids from potential bidders that are screened out
may not be observed. Second, data may only include auctions with at least one bid above
r, and exclude those where everyone is screened out (i.e. X (1) < x (r)). In both cases, the
algorithm in our benchmark model above can not be applied immediately.
In addition, a binding reserve price r in data also reduces the scope of counterfactual
reserve prices that are interesting for counterfactual analyses. To understand this, note
28
bids below r reveal no information about underlying signals, as the link between GrB and
structures 2
F only holds for br (x; ) r. The data at hand cannot help address the
question how those bidders who only become unscreened under a lower counterfactual price
will act. Hence the logic behind the bounds in our benchmark case only applies to revenue
distributions in counterfactual auctions with r0 > r. As a result, for all r0 < r, x (r0 ) is lower
than x (r) and can not be bounded in its small neighborhoods using equilibrium conditions.
Throughout this subsection, I focus on the bounds for FRI (r) . Extensions to bound FRII (r)
is straightforward and omitted.
5.2.1
Unobserved screened bidders
Unobserved bids from bidders who are screened out matter for bounding FRI (r0 ) (where
r0 > r) only in the sense that they may make the number of potential bidders unobservable.
For now assume auctions with X (1) < x (r) are also observed in the data. If the number
of potential bidders is known, as is often the case in applications, then the algorithm for
bounding FRI (r) can be applied even if data do not contain bids from bidders that are screened
out. The following lemma generalizes the equilibrium condition (2) for any rationalizable
distributions under a binding reserve price r.
Proposition 7 Consider any distribution of bids GrB in …rst-price auctions with a binding
reserve price r. Then for all 2 (GB0 ), (br (x); GrB ) = vh (x; ) for all x x (r; ).
In the presence of binding reserve prices in data, the lower bound on v(x; x; ) can no
longer be identi…ed from GrB , as bids lower than r can not be linked to signals through
equilibrium conditions. The solution is to bound v(x; x; ) below by expected payment of a
winner in second-price auctions with a reserve price r. For x x (r; ) de…ne
Z x
FY jX (x (r)jx)
f
(sjx)
vl;r (x; ) r FY jX (xjx) +
vh (s; s; ) FYY jX
ds
jX (xjx)
x (r; )
Then vl;r (x; ) is increasing in x by monotonicity of the value function and a¢ liations between signals, and v(x; x; )
vl;r (x; ) for x
x (r; ) by the equilibrium condition in
second-price auctions with r. (The formal proof is similar to the benchmark case and omitted.) Hence for all r0 > r, x (r0 ; ) is bounded by xh;r (r0 ; ) arg minx2[x (r; );xU ] (vl;r (x; )
r0 )2 and xl;r (r0 ; ) arg minx2[x (r; );xU ] (vh (x; x; ) r0 )2 . Then vh (x; x; ) and vl;r (x; ) are
identi…ed from GrB for x x (r; ) respectively as (br (x); GrB ) and
Z br (x)
GrM jB (rjbr (x))
gr
(rjbr (x))
r
r
r Gr (br (x)jbr (x)) +
(~b; GrB ) Gr M jB(br (x)jbr (x)) d~b
l;r (b (x); GB )
M jB
r
M jB
29
By similar reasoning as in the benchmark case, bounds on the
br (x) into br (x) for x x (r0 ; )) are identi…ed as
r;r 0 ;k (b
r
(x); GrB )
=r
0~
L(brk;r0 jb; GrB )
+
Z
b
r
r0 -mapping
(which maps
~ ~bjb; GrB )
(~b; GrB )dL(
brk;r0
where brk;r0
br (xk;r (r0 ; )) for k = l; h, and are identi…ed as inverses of (:; GrB ) and
r
r
r
r
l;r (:; GB ) over [r; b (xU )] respectively. It can be shown that r;r 0 ;k (b (:); GB ) is increasing
0
for x xk;r (r0 ), and inverting r;r0 ;k (:; GrB ) at t r0 gives bounds on br ( r (t)). Thus bounds
on FRI (r0 ) can be constructed from the distribution of br (X (1) ).
5.2.2
Unobserved screened auctions (with X (1) < x (r))
When data exclude auctions with a reserve price r that screens out all bidders (i.e. X (1) <
x (r)), we observe the distribution of equilibrium bids br conditional on more than one bidder
bids above r (denoted GrBjB (1) r ) as opposed to the unconditional GrB . For b > r, GrM jB (bjb)
r
r
0
and gM
jB (bjb) can still be identi…ed from GBjB (1) >r , and thus bounds on br (x (r )) and the
r
r;r 0 -mapping can be constructed as above. However, GBjB (1) r can only be used to construct
bounds on FRI (r)jX (1) r . That is, for any rationalizable GrB and 2 (GrB ),
Pr( (Br(1) ; GrB ) < r0 jBr(1)
Pr(X
0
< x (r )jX
(1)
r
l;r (Br ; GB )
(1)
r)
x (r))
< r0 jBr(1)
r)
Pr(Br(1)
1
r
(1)
r;r 0 ;l (t; GB )jBr
r)
Pr(X (1)
r0
Pr(Br(1)
1
r
(1)
r;r 0 ;h (t; GB )jBr
Pr(
and for t
(1)
r0 ,
(t)jX (1)
x (r))
r)
(1)
where Br is shorthand for br (X (1) ). The probability that r screens out all bidders Pr(X (1) <
x (r)) is needed to bound the unconditional distribution FRI (r) . It is impossible to identify
this probability solely from GrBjB (1) r without further restrictions on FX . However, the lemma
below shows when bidder signals are i.i.d., Pr(X (1) < x (r)) can be recovered from GrBjB (1) r
alone. 23
23
Under A1, the auction model still has interdependent values even when fXi gi2N are independent and
identically distributed.
30
Proposition 8 Suppose signals fXi gi=1;::N are i.i.d. in …rst-price auctions with N potential
bidders and a binding reservation price r. If both the number of active bidders and N are
observed, then Pr(X (1) < x (r)) is identi…ed even if auctions with X (1) < x (r) are not
observed.
5.2.3
About the number of potential bidders
That the number of potential bidders N is observed is crucial to our discussion of data
generated under binding reserve prices so far. This is not an issue in some applications
where N is directly reported in the data, or where good proxies exist. In other applications,
the issue can be subtle. In some cases, neither bidders nor econometricians can observe
N . Then strategic decisions can be modeled as based on bidders’ subjective probability
distributions of N given private signals (denoted p(N = njX = x)).
Bidders integrate vh;N , fY jX;N over N with respect to this distribution and make strategic decisions based on these integrated primitives, so the actual number of potential bidders
becomes irrelevant in equilibria. The new equilibrium conditions can also be manipulated
through change of variables to get an analog of (2) that links bid distributions observed
to model primitives. One example is the o¤-continental shelf (OCS) auctions of oil-drilling
rights studied by Hendricks, Pinkse and Porter (2003). In OCS auctions, potential bidders’
decisions to submit bids take multi-stages. The authors endogenize participations by introducing multiple signals, each corresponding to a stage in the decision-making. Then only
those still active in the last-stage and their signals are relevant to decisions on strategic
bids. The additional restrictions in the model is that decisions to remain active till the
last stage only depends on signals from previous stages, and that conditional on last-stage
signals, the signals in previous stages reveal no information about bidders’values. The logic
of partial identi…cation in benchmark models can be extended in principle to bound revenue
distribution in such equilibria with unobserved potential bidders.
In other applications where bidder signals are i.i.d., the number of potential bidders can
be identi…ed even if data only report the number of actual bidders. This is because in
equilibria, the number of actual bidders is distributed as Binomial (n; p) with p equal to the
screening probability Pr(x
x (r)). Provided the distribution of bids and actual bidders
are rationalizable,24 both n and p are uniquely identi…ed.
24
See Guerre et.al (2000) for conditions for rationalizability.
31
6
Application: U.S. Municipal Bond Auctions
Municipal bonds are a chief means of debt-…nancing for U.S. state and county governments.
They are issued to …nance public projects such as construction or renovation of schools
and public transportation facilities. Interest income from municipal bonds are exempt from
federal and local taxes, and hence municipal bonds appeal to investors in high tax brackets.
In 2005, the total par amount of outstanding municipal bonds was $1.8 trillion. 25
6.1
Institutional details
Muni-bonds are identi…ed by issuers and basic features such as coupon rates, maturity dates,
and par amounts.26 Investors valuate muni-bonds based on this information and implied
risks, including credit risks, interest rate risks, and liquidity risks.27 On the primary market,
muni-bonds are issued through …rst-price auctions to potential underwriters (mostly investment banks). Notices of these competitive sales are posted on major industry publications
such as The Bondbuyer. In practice, issuers usually package a series of bonds for sale in one
auction, and investment banks participate by bidding a single dollar price per $100 par value
for the whole series. The bidder with the highest dollar price wins the right to underwrite
the entire series, and may resell the series on secondary markets with a mark-up. To decide
whether and how to bid, securities …rms assess the creditworthiness of the municipalities
and prospects of the bonds on secondary markets. For issues with a large par amount, investment banks usually form bidding syndicates, where members share responsibilities for
reselling the bonds as well as the liability for unsold bonds. A syndicate is usually clearly
de…ned for each issuance, as underwriters traditionally stay in the group where they bid on
the last occasion that the issuer came to market. As of 2006, more than 2,100 securities
…rms are registered with the Municipal Securities Regulatory Board and authorized as legal
underwriters. However, only a small number of these …rms are active bidders in competitive
sales. By 1990, 25 leading underwriters managed about 75 percent of the total volume of all
25
Source of information : SIFMA(2005)
A coupon rate is the interest rate stated on the bond and payable to the bondholder on a semi-annual
basis. A maturity date is the date on which the bondholder will receive par value of the bond along with its
…nal interest payment.
27
Credit risk measures how likely the issuer is to default on its payment of interests and principals. Interest
rate risk is due to ‡ucuations in real interest rates that a¤ect the market value of bonds (to both speculators
and long-term investors). Liquidity risk refers to the situation where investors have di¢ culty …nding buyers
when they want to sell, and are forced to sell at a signi…cant discount to market value.
26
32
new long-term issues either as lone bidders or leaders of syndicates.
6.2
Bond values: private or common ?
The bounds proposed introduce an approach of partial identi…cation for policy analyses which
is applicable regardless of underlying paradigms (PV or CV ). This is highly relevant in the
context of muni-bond auctions, as institutional details do not suggest conclusive evidence
for either paradigm. Besides, empirical methods proposed so far for di¤erentiating the two
all have limitations in practice.
The value of bonds for …rms in these auctions are resale prices on secondary markets.
On most occasions bidders on the primary market cannot foresee at what price they can
resell the bonds, and therefore only have noisy estimates. These estimates capture the
syndicates’expectation on how investors on secondary markets interpret bond features, and
depend on their beliefs about the skills of their sales and trading sta¤. The estimates are
also built on companies’perception of how investors view relevant uncertainties such as the
creditworthiness of municipalities and ‡uctuations of future real interest rates.
The crucial question is whether a bidding syndicate can extract additional useful information about bond values if they could access competitors’estimates. The auction is one
with common values if and only if the answer is positive. On some occasions, all participating …rms manage to pre-sell bonds to secondary investors prior to their actual bidding. Such
auctions …t in the PV paradigm, as all bidders have perfect foresight of their values. On
other occasions, pre-sales are not possible or limited in scope, and …rms can have heterogenous source of information about municipalities’creditworthiness, or di¤erent interpretation
of factors related to bond values. Unless all …rms con…dently believe their own information
or interpretation dominates their competitors’in accuracy, they cannot dismiss competitors’
estimates as uninformative, and auctions are closer to common values.
While the informational environment per se does not justify either P V or CV conclusively,
data limitations also deter empirical e¤orts to discriminate between them. First, there is
reason to believe the number of potential bidders is correlated with bond values. Therefore
the test in Haile, Hong and Shum (2003) cannot be applied, for it requires the variation in
the number of bidders to be exogenous with respect to the distribution of values. Second,
our data does not have ex post measures of bond values that can be used to test whether
vh (x; x) = E(Vi jBi = b0 (xi ); B i = b0 (x i )) is independent from B i . Finally muni-bond
auctions proceed with no announced reserve prices and therefore the testable restrictions in
33
Hendricks, Pinkse and Porter (2003) are not useful.
This paper focuses on an robust approach for policy analyses by bounding counterfactual
revenue distributions under general restrictions that encompass both PV and CV paradigms.
The lower bound point-identi…es counterfactual distributions only when values are private.
On the other hand, if nothing is known about the interdependence between values, then any
point between the bounds can be rationalized as the true counterfactual revenue distribution
by certain structures in the identi…ed set de…ned by the observed bid distribution.
6.3
Data description
The data contains all bids submitted in 6,721 auctions of municipal bonds on the primary
market in the United States between 2004 and 2006. They are downloaded from auction
worksheets at a website of Thompson Financial. The data reports the identity of issuers, the
sale date, the date of the …rst coupon, par values of each bond in a series, coupon rates of
each bond, S&P and Moody’s ratings of each bond, the type of government credit support for
the issuance (general obligation or revenue).28 It also records whether the issuance is bankquali…ed.29 In addition, the data includes macroeconomic variables measuring opportunity
costs of investing in bonds.
There are 97,936 bonds in 6,721 series, with an average of 14.5 for each issue. About 70%
of the series have 10 to 20 bonds. The average coupon rate of all bonds is 4.06% and the
average number of semiannual payments is 19.6. I use the par-weighted averages of coupon
rates and numbers of coupon payments as a measure of "overall" interest rates and maturity
for a series. About 90% of all issuances have a weighted average coupon rate between 3%
and 5%. The weighted average maturity is approximately normally distributed with a mean
of 20.8 and a standard deviation of 9.5. The total par of a series ranges from $0.1 million
to $809 million, and is skewed to the right with a mean of $21.4 million and a median of $6
million. About 64.5% of the series are backed by full credit of municipalities, while the rest
28
Bonds are categorized into two groups by the degree of credit support from municipalities. General
obligation bonds are endorsed by the full faith and credit of the issuer, whereas revenue bonds promise
repayment from a speci…ed stream of future income, such as that generated by the public project …nanced
by the issue. The latter usually bears higher interest rates due to risk premium.
29
The Tax Reform Act of 1986 eliminated the tax bene…ts for commercial banks from holding municipal
bonds in general. But exceptions were made for "bank-quali…ed" bonds, for which commercial banks can
still accrue interests that are tax-exempt. Hence banks have a strong appetite for bank quali…ed bonds that
are in limited supply, and bank quali…ed bonds carry a lower rate than non-bank quali…ed bonds.
34
are backed by limited municipal support, such as revenue stream from public works …nanced
by the issuance. In practice, issuers have the option to include reserve prices in the notice
of sale, but few issuers use this option. For each auction, the number of bidding coalitions,
the number of companies within each coalition and their identities are all reported in the
data. The number of syndicates ranges from 1 to 20, with a mean of 5.6 and a standard
deviation of 2.6. Series that received more than 3 but fewer than 7 bids account for 68% of
all auctions.
The dollar prices tendered are not always reported. However, total interest costs for all
bids are always reported.30 I use the following formula to calculate and impute missing dollar
bids :
PQ PTq 1 Cq =2
Pq
q=1
t=0 (1+ T IC )t + (1+ T IC )Tq
tf
2
2
100
B = (1 + T IC)
PQ
q=1 Pq
where q indexes bonds in a series of Q bonds, Tq is the number of semi-annual periods
from the date of …rst coupon until maturity, Cq and Pq are coupon and principal payments
respectively, tf is the time until …rst coupon payment and B is the dollar bid per $100 of face
value. Table 1 summarizes the distribution of all 37,547 bids submitted in 6,721 auctions.
The 1st percentile is $95.32 and the 99th percentile is $109.30. The median is $99.40, the
mean is $99.92, and the standard deviation is $2.76. The median winning dollar bid is $99.66,
the average is $100.01, and the standard deviation is $2.36.
6.4
Homogenization of bids
There is a wide variation of bond features in the data. In competitive sales, syndicates take
these characteristics into account when they bid, and thus strategies across auctions are not
homogenous as the benchmark model posits. In principle bounds still apply to subsets of
homogenous auctions where bond features are controlled. The main empirical challenge in
the implementation is that constructing nonparametric bounds on conditional revenue distributions require large samples for auctions with …xed speci…c features. Below I tackle this
issue by homogenizing bids across heterogenous auctions. The working assumptions are: (i)
…rms’estimates of bond values are independent from publicly known bond features conditional on the number of participating syndicates; (ii) value functions are additively separable
in private signals and commonly observed bond features. Under these assumptions, marginal
30
Total interest cost (TIC) is the interest rate that equates dollar prices with discounted present value of
future cash‡ows the series.
35
e¤ects of bond characteristics on equilibrium bids are identi…ed. (I discuss a speci…cation test
of these restrictions below.) Thus bids in distinct auctions can be homogenized by removing
di¤erences due to variations in bond features as in Section 5.
In competitive sales with n bidding syndicates, ex ante bond values for a potential bidder
is :
Vil = Z0l +
n (Xil ; X il )
where i = 1; ::; n indexes the bidding syndicates, l = 1; 2; ::Ln indexes auctions with n syndicates, Zl is a vector of publicly known features, and Xl = (Xil ; X il ) is a Rn -valued random
vector of idiosyncratic signals. This speci…cation re‡ects the intuition that marginal e¤ects
of idiosyncratic information (signals Xl ) may not interact with those of public information
(bond features Zl ). Syndicates in an auction may di¤er in two aspects: the number of member …rms, and local presence of …rms’branch o¢ ces in the issuer’s state. Recent empirical
works suggest there is no conclusive evidence that they can lead to informational asymmetries.31 Hence I maintain symmetry restrictions of n and FX as in the benchmark model in
Section 2.
The equilibrium strategy is:
bil (xil ; zl ; n) = z0l +
l (xil ; n)
(4)
Rx
Rx f
(uju)
dug and n (s) E[ N (Xi ; X i )
where (x; n)
(s)dLn (sjx), Ln (sjx) expf s FYY jX;n
xL n
jX;n (uju)
j Xi = maxj6=i Xj = s; N = n]. Thus strategic bids can be decomposed into two additive
components. The …rst term suggests marginal e¤ects of bond features are invariant to potential competitions, and the second term captures e¤ects of potential competition on strategic
bids. The signals and competitions interact with each other and their e¤ects cannot be
separated. Regressing bids on bond features and a vector of dummies for the number of
potential bidders will estimate consistently. That is, in the pooled regression,
bil (xil ; zl ) = d0l + z0l + uil
(5)
where dl is a vector of dummies for n, the error term uil is mean independent conditional
on dl and zl .32
31
32
See Shneyerov (2006).
To see this, …x n, then Proposition 5 shows equilibrium bids are:
bil (n) =
0 (n)
+ z0l + "il (xil ; n)
where 0 (n) E[ l (Xil ; Nl )jNl = n] and "il (xil ; n)
l (xil ; n)
0 (n). It follows from the independence of
Xl and Zl conditional on number of bidders that E["il (Xil ; Nl )jNl = n; Zl = zl ] = 0 for all (nl ; zl ).
36
6.4.1
GLS estimates of index coe¢ cients
When there is intracluster correlation among error terms within auctions, a simple ordinary
least square estimator will be ine¢ cient. This can happen when syndicates’ signals Xl
are strictly a¢ liated. One explanation for a¢ liated signals in the …nance literature is the
"herding" e¤ect among research and sales sta¤ across syndicates. For example, researchers
in di¤erent syndicates tend to have similar professional backgrounds or trainings and hence
are inclined to make similar decisions on the choice and weights of value-related factors in
their analyses. Strict a¢ liation among signals could also happen when syndicates’estimates
consist of idiosyncratic noisy measurements of a common, underlying random variable.
Table 2 below reports the GLS estimates and t-statistics of for equation (5). The
dependent variable is the dollar price bid. The regressors include publicly known bond
features : weighted average coupon rate (wacr), weighted average maturity (wapn), total
par value of the series (totpar), a dummy for whether the series is supported by full municipal
credit (sectype), a dummy for whether the series is bank-quali…ed (BQ), a dummy for whether
the series is rated with investment grade (HR) and two interaction terms type_cr and HR_pn
respectively.33 Butler (2007) suggests local presence of syndicates in the geographical area
of the issuer could also in‡uence their private information about the credibility of the issuer
and hence their estimates of the value of the series. Therefore I also include in the regressors
some dummies for the regions, MW (Midwest), NE (New England), SW (Southwest), South
and West, to test the impact of geographic location on bids.
The weighted average coupon rates and maturity are both highly signi…cant at 1% level,
with positive and negative marginal e¤ects respectively. These estimates con…rm the intuition that bond values increase with cash‡ows from coupons and decrease as maturity
increases because of higher risks in the ‡uctuation of interest rate and in‡ation. Municipality support has a signi…cant positive e¤ect on the bids. Controlling for other features, the
average dollar price is $2.47 higher for bonds supported by the full credit of municipalities.
Bond ratings by S&P and Moody’s have no signi…cant impact on bids ceteris paribus. A
possible explanation is that the syndicates’research forces do not consider ratings informative conditional on their own research on bond values. The dollar bid for bank-quali…ed
series are on average about 84 cents lower than non bank-quali…ed ones. The e¤ect is statistically signi…cant at 1% level. Besides, an increase of $1 million in total par leads to a slight
increase of 1.76 cents in the dollar price. This can be explained by the fact that average
participation costs for a syndicate (e.g. time and e¤ort on research) per $100 in par is lower
33
The unit for wapn is 10 semin-annual coupon payments and the unit for totpar is $100 million.
37
for issuance with larger par amount. The interaction of sectype and wacr are also highly
signi…cant at 1% level, suggesting marginal e¤ects of coupon rates are lower for series with
full municipal credit supports. There is no conclusive evidence for regional e¤ects on bids
except that dollar prices for series issued in New England are higher on average than those
issued in the Midwest.
6.4.2
Speci…cation tests
Two identifying restrictions in the regression equation (5) are additive separability and conditional independence of bond features and signals in value functions. A testable implication
of these two restrictions is that marginal e¤ects are constant and invariant to the number of
potential bidders. That is, for each n, the following regression equation holds:
bil (n) =
0 (n)
+ z0l + "il (xil ; n)
where 0 (n) E[ l (Xil ; Nl )jNl = n] and "il (xil ; n)
l (xil ; n)
0 (n) is mean-independent
conditional on Zl and n. On the other hand, if either restriction is not satis…ed, bidding
strategies are nonseparable in Zl , Xil and n. Consequently, marginal e¤ects of bond features on bids change with the number of potential bidders. Therefore we can test the two
restrictions jointly by comparing estimates for auctions with di¤erent numbers of bidding
syndicates.
Table 3(a) reports GLS estimates in regressions for n between 4 and 8. The choice
of regressors z is the same as that in (5). The estimates are consistent across n in signs
and signi…cance. For each signi…cant characteristic of the series, Table 3(b) reports test
statistics for the pair-wise hypotheses that coe¢ cients are the same in the two regressions
with di¤erent n. The statistics are constructed as the ratio of di¤erences between GLS
estimates and the standard error of the di¤erence.34 Under null hypotheses, the test statistics
are asymptotically standard normal.
The results show di¤erences between sizes of estimates are insigni…cant. With the exception of weighted average coupon rates for n = 4, all other estimates are not signi…cantly
di¤erent from their counterparts under a di¤erent n. There is no statistically signi…cant
evidence against the hypotheses that the value function is additively separable and bond
34
Note GLS estimators for di¤erent n are independent, for (Zl ; Nl ; Xl ) are i.i.d. draws from the same joint
distribution. Hence the standard deviation of the di¤erence in two estimators can be consistently estimated
by adding up their standard errors.
38
features have no bearing on the distribution of idiosyncratic signals conditional on the number of participating syndicates.
6.5
6.5.1
Results
Point and interval estimates for F^Rk I (r) and F^Rk II (r)
This section reports bound estimates on counterfactual revenue distributions for a reference
bond series when there are n = 4 bidding syndicates in the auction. The reference series is
issued in the Midwest, bank-quali…ed, backed by full municipal credit, and has an investment
grade from S&P and the Moody’s. The reference series has a weighted average coupon rate
of 4% and maturity of 5 years, as well as a total par of $4:84 million. These are the medians
of features of the bond series among auctions with 4 bidding syndicates.
Figure 5(a) plots kernel density estimates of the ordered bids that are "homogenized" at
the reference level, which are calculated using GLS estimates in regressions with 4 bidders.
Distributions of the ordered bids are approximately normally distributed with similar standard deviations and the di¤erences between the median of adjacent ordered bids are between
$0.25 and $0.35 per $100 in par amount. I use the product of tri-weight kernels for estimating
GM;B and gM;B . The choice of bandwidths follows the "rule of thumb" discussed in Monte
Carlo section.35 The data is parse close to the both boundaries even after trimming bids
that are within one bandwidth from the minimum and maximum bids reported. To avoid
poor performances of the kernel estimates of ^l for lower dollar values, I trim the bids at the
0.5-th and 99.5-th percentile.36 In the data, bids from the same auction are almost always
trimmed together.
Figure 6 plots estimates ^ and ^l and suggests the distance between estimates of bounds
on b0 (x (r)) only widens slowly as r increases. That ^l stays mostly above the 45-degree
line is evidence for strict a¢ liations between private estimates within each auction. Table 4
below summarizes estimated bounds on b0 (x (r)) and the probability that no one bids above
r (hereafter referred to as the all-screening probability) for di¤erent reserve prices.
35
1
The bandwidths hG and hg are respectively 2:98 1:06^ b (4L4 ) 4n 5 = 2:43 and 2:98 1:06^ b
1
(4L4 ) 4n 4 = 2:57.
36
The distance between the minimum bid and the 0.5-th percentile is about $5. The number is greater
than the smoothing parameter hg = 2:57 used in the estimation.
39
Table 4 : Estimated bounds on the all-screening probability
r
98
99
100
101
102
103
^b0 (xl (r)) ^b0 (xh (r)) b:w: of b0 (x (r)) F^ l I (r ) F^ uI (r )
R (r)
R (r)
97:17
98:00
98:76
99:45
100:13
100:74
97:89
98:83
99:73
100:60
101:39
102:14
0:72
0:83
0:97
1:15
1:26
1:40
0:0540
0:1488
0:3609
0:5935
0:8074
0:9042
0:1256
0:3860
0:6865
0:8837
0:9516
0:9702
Table 4 suggests marginal bidders under r are estimated to bid lower than r in the scenario
with no binding reserve price. It is consistent with the theoretical prediction that FRI (r) (r) is
less than FRI (0) (r). The di¤erence between the boundwidths of the all-screening probability
for r = 98 and r = 100 is mostly due to the distribution of winning bids with no binding
(1:4)
reserve prices. Figure 5(b) shows the distribution of b0 (the winning bid out of 4 bids) has
a larger mass in [b0 (xl (100)) b0 (xh (100))] = [98:75 99:73] than in [b0 (xl (98)) b0 (xh (98))] =
[97:17 97:89]. Therefore the bounds on the all-screening probability is much wider for
r = 100 even though bounds on b0 (x (100)) is only slightly wider than those of b0 (x (98)).
For reserve prices between $98 and $103, the solid and dotted lines in the panels of Figure
7 depict point estimates F^RuI (r) and F^Rl I (r) respectively. In addition, I construct 100 bootstrap
samples, each containing 1075 auctions drawn with replacement from the estimating data.
For all levels of revenue, I record the 5-th percentile of F^Rl I (r) and 95-th percentile of F^RuI (r) .
They form a conservative, pointwise 90% con…dence interval of [FRl I (r) ; FRuI (r) ], and are plotted
in Figure 7 as broken lines. In addition, the table below reports the bounds on major
percentiles according to the estimates of bounds on F^Rl I (r) and F^RuI (r) .
Table 5 : Estimated bounds on quartiles of FRI (r)
r
98
99
100
101
l:b: 1st
98:47
99:08
v0
v0
u:b: 1st
98:66
99:30
v0
v0
l:b: 2nd
99:18
99:29
v0
v0
u:b: 2nd
99:28
99:54
100:07
v0
l:b: 3rd
99:93
100:01
100:14
v0
u:b: 3rd
99:98
100:14
100:48
101:09
Revenue distribution above the reserve price depends on the distribution of b0 (x) and the
0
r functional mapping b0 (x) and GB into br (x). The densities plotted in Figure 5 (a) illustrate
40
homogenized winning bids are approximately normally distributed. Besides, our estimates
of bounds on r are approximately linear. Therefore bound estimates F^Rk I (r) (t) for t > r
increase at decreasing rates, a pattern similar to normal distributions. By construction,
estimates of bounds on the all-screening probabilities are monotone in reserve prices (i.e.
F^Rk I (r) (r ) is increasing in r for k = l; u). In addition, our estimates suggest that for any pair
of reserve prices r < r0 , F^Rk I (r0 ) (t) < F^Rk I (r) (t) for t r0 . This is consistent with the theoretical
prediction that for a given signal above the screening level, …rms bid less aggressively when
the reserve price is lowered. Likewise Figure 8 plots point estimates for revenue distribution
in second-price auctions as well as the 90% con…dence intervals for [FRl II (r) (t); FRuII (r) (t)].
6.5.2
Choice of optimal reserve prices
Knowledge of revenue distributions in counterfactual auctions makes it possible to use other
distribution-based criteria for comparing auction revenues, instead of expectations alone.37
This is especially useful when the seller is known to be risk-averse and expected utilities are
used as criteria.
A natural consequence of our partial approach is that only bounds on these criteria
functions can be calculated. Such bounds on criteria functions are also tight and exhaust
all information possible from equilibrium bids without further restrictions on value functions
and signal distributions. As a result, answers to policy questions above involves comparing
bound estimates rather than point estimates. Bounds on criteria functions can also be used
to bound optimal reserve prices following the logic in Haile and Tamer (2003).
A value for v0 is needed for calculating both upper and lower bounds on E(RI (r)) and
E(RII (r)). This should be measured by the amount of money that a municipality would be
able to raise if it had borrowed through an alternative, next-cheapest channel (i.e. a creditor
that requires the next lowest interests than syndicates in the auctions). The proxy for v0
I use here is $95:71, and it is calculated as the present value per $100 in par of cash ‡ows
from the coupon and principal payments of a reference bond, with the discount rate being
the 99-th percentile of total interest rates reported in the data.
Figure 9(a) plots estimated upper and lower bounds on E(RI (r)) (denoted E^h (RI (r)) and
E^l (RI (r)) respectively), which are calculated from F^Rl I (r) and F^RuI (r) through discretization
and numerical integration using midpoint approximations. The solid lines plot E^k (RI (r))
37
Within …rst-price auctions, each r > v0 can be justi…ed as optimal under the criterion of maximizing
Pr(RI (~
r) r). That is r = arg maxr~>v0 Pr(RI (~
r) r) for all r > v0 .
41
and the dotted lines plot E^k (RII (r)). The upper bounds of expected revenue correspond to
the case of P V auctions. Note estimates for E^h (RII (r)) are higher than E^h (RI (r)) for almost
all r in the range. This is consistent with the implication of Revenue Ranking Principle:
for a …xed level of r, the expected revenue is higher for second-price auctions when signals
are a¢ liated. For …rst-price auctions, E^h (RI (r)) is maximized at r = $98:68 to be $99:29,
and E^l (RI (r)) is maximized at r = $96:26 to be $99:16. An argument similar to Haile
and Tamer (2003) suggests the optimal reserve price that maximizes E(RI (r)) must be in
the range [$96:12; $99:21]. For second-price auctions, E^l (RI (r)) and E^h (RI (r)) are both
maximized at r = $96:57 with the maximum $99:94, thus providing a point estimate for
E(RI (r))-maximizing reserve price. Instead of calculating a range of r that maximizes the
expected revenue, an alternative is to pick r that maximizes either the lower or upper bound
on E(RI (r)). In the case of risk-neutral bidders, estimates for E^l (RII (r)), E^h (RII (r)) and
E^l (RI (r)) are all close to being monotone, and their maximizers are all close to the boundary
$96.
A major motivation for focusing on revenue distribution in counterfactual analyses is the
risk aversion of the seller. Given any speci…cation of the seller’s utility function (denoted
u(t)), fF^Rk j (r)gk=l;u
j=I;II can be used to estimate bounds on the seller’s expected utility (denoted
k=l;u
fUk (FRj (r) )gj=I;II ). Like the case with a risk-neutral seller, these bounds can be used to
put a range on an optimal reserve price that maximizes U (FRj (r) ), or be used as criteria
themselves for choosing reserve prices.
I consider three speci…cations of the seller’s utility function: uDARA (t) = ln(t) (DARA)
1
and uCRRA (t) = t1 with = 0:6 and 0:9 (CRRA). Figure 9(b), (c) and (d) plot estimated
bounds on the expected utilities in …rst- and second-price auctions (i.e. fUk (FRj (r) )gk=l;u
j=I;II )
for DARA, CRRA( = 0:6) and CRRA( = 0:9) utility functions respectively. Table 6
below summarizes reserve prices that maximize estimated bounds of expected utilities in
…rst-price auctions, as well as estimated bounds on optimal r maximizing expected utilities.
Table 6 : Optimal reserve prices for …rst-price auctions
r(maximizer)
maximum
bounds on r
U^lDARA U^hDARA
96:19
98:65
4:593
4:594
[96:24; 99:20]
U^l =0:6 U^h =0:6 U^l =0:9 U^h =0:9
96:23 98:68 96:25 98:66
15:711 15:719 15:822 15:824
[96:17; 99:32]
[96:21; 99:20]
In second-price auctions, estimates of bounds on expected utilities under di¤erent speci…cations are all maximized at $96:26, with the maxima being 4:594, 15:743 and 15:808
42
respectively. As a result, we get a point estimate of the optimal reserve price r at $96:26 for
all three speci…cations. Both maximizers across di¤erent speci…cations of utility functions
are close to each other and so are the interval estimates. This is because uDARA , u =0:6 and
u =0:9 are all approximately linear for the range of revenues considered in this application.
As a result, estimated bounds on fU (FRj (r) )gj=I;II as functions of r are close to being linear
transformations of each other.
On the other hand, estimates for di¤erent u(:) yield di¤erent implications regarding the
choice of format between …rst- and second-price auctions. For DARA utility functions, the
point estimate for the optimal reserve price in second-price auctions is $96:26, with a maximum U^ DARA (FRII (96:26) ) = 4:594. This is equal to the maximized value for U^hDARA (FRI (98:65) ).
Hence estimates suggests a seller with decreasing absolute risk aversion should prefer secondprice auctions in general, and may be indi¤erent between the two formats if the auction is
known to belong to the P V paradigm. For CRRA utilities with = 0:6, the implication is
the same as in the case with risk-neutral sellers. However, for CRRA utilities with = 0:9,
estimates suggest …rst-price auctions should be preferred over second-price ones. The pattern
is due to the fact that FRI (r) always crosses FRII (r) from below for any given r, and u =0:6
increases faster than u =0:9 .
Finally a technical note is in order. Except for E^h (RI (r)) and U^h (RI (r)), other estimates
of bounds on fE(Rj (r))gj=I;II and fU (Rj (r))gj=I;II are almost monotonically decreasing in
r. In general this need not be the case in estimation. To see this, note that none of the
estimates fF^ k (Rj (r))gk=l;h
j=I;II reported in Figure 7 and Figure 8 are stochastically ordered in
r. In this incidence, the monotonicity is explained by the fact that our measure of v0 is low
at $95:71 and that estimates ^br (xh (r0 )) are close to r0 for all (r; r0 ).
7
Conclusion
In structural models of …rst-price auctions, interdependence of bidders’values leads to nonidenti…cation of model primitives. That is, distributions of equilibrium bids observed in a
given auction format can be rationalized by more than one possible speci…cations of signal
and value distributions. While this negative identi…cation result rules out policy analyses
that rely on exact knowledge of primitives, the distribution of bids observed in equilibria
should still convey useful information about primitives that can be extracted for counterfactual revenue analyses. This paper derives bounds on revenue distributions in counterfactual
auctions with binding reserve prices. The bounds are the tightest possible under restrictions
43
of interdependent values and a¢ liated signals, and can be used to compare auction formats
or bounds on optimal reserve prices. This approach also addresses the empirical di¢ culty of
di¤erentiating P V and CV paradigms in policy analyses. The bounds can be nonparametrically consistently estimated, and Monte Carlo evidence suggests these estimators also have
reasonable …nite sample performances.
Observed heterogeneity in auction characteristics can be controlled for by conditioning
counterfactual analyses on these auction features. Under the restriction of additive separability of signals and auction characteristics in value functions, the marginal e¤ects of auction
features can be identi…ed if signals are independent from auction features conditional on
the number of bidders. By removing variations due to observable auction heterogeneity, the
bids across various auctions can be "homogenized" to bids in auctions with given speci…c
features. The issue of data generated under a binding reserve price also does not pose major
challenges to the construction of bounds, provided the data report the number of potential
bidders or good proxies of this number.
Applying this methodology to U.S. municipal bond auctions on the primary market
yields informative bound estimates of revenue distributions in counterfactual auctions with
binding reserve prices. These estimates are then used to bound the reserve prices that
maximize expected revenues for risk-neutral sellers. For risk-averse sellers, bounds on revenue
distributions are also used to bound optimal reserve prices which maximize their expected
utility under di¤erent speci…cations of utility functions.
An interesting direction for future research include extensions of partial-identi…cation
methods for more complicated cases such as asymmetric information among bidders and
unobserved auction heterogeneity. Another promising direction is inference using our threestep bound estimates. For example, suppose data reports exogenous variations in binding
reserve prices. Then a novel test for private values can be constructed by comparing the
actual revenue distribution under a higher reserve price and the hypothetical upper bound
on the revenue distribution constructed from bids in auctions with lower reserve prices.
44
8
Appendix A: Proof of identi…cation results
Proof of Proposition 1. To prove necessity, suppose 2 CV F generates G0B in such
an equilibrium. Then the support of B is SBN [b0L ; b0U ]N , with b0L
vh (xL ; xL ; ) and
0
0
0 N
bU = b0 (xU ; xU ; ). Note 8b 2 [bL ; bU ] ,
GB0 (b)
Pr(b0 (X1 ; )
= Pr(X1
FX (
0
0
b1 ; ::; b0 (XN ; )
(b1 ; ); ::; XN
0
bN )
(bN ; ))
(b; ))
where 0 (:; ) denotes the inverse bidding strategy in equilibrium under the structure and
no binding reserve prices. Symmetric equilibrium and exchangeability of FX implies GB0 (b)
must be exchangeable in b for all b 2 SBN . The a¢ liation of B = (b0 (X1 ; ); ::; b0 (Xn ; ))
follows from the monotonicity of b0 (:) and the a¢ liation of X by Theorem 3 in Milgrom
and Weber (1982). The …rst-order condition (2) implies (b; GB0 ) = vh ( 0 (b; ); 0 (b; ); )
8b 2 [b0L ; b0U ], where vh (x; x; ) is increasing in x on SX by the de…nition of CV F. Hence
the strict monotonicity of 0 (b; ) implies (b; GB0 ) is increasing over SB . The proof of
su¢ ciency uses a claim and an example below.
Claim A1 Suppose a rationalizable bid distribution GB0 that satis…es conditions (i) and
(ii) in Proposition 1. Then a structure = ( ; FX ) 2
F rationalizes GB0 if and only
N
satis…es the following …xed-point equation for all x 2 SX ,
FX (x) = GB0 (
1
(vh (x1 ; x1 ; ); GB0 ); ::;
1
(vh (xN ; xN ; ); GB0 ))
(6)
Proof of Claim A1 Suppose 2
F rationalizes such a GB0 in an increasing, symmetric
N
equilibrium. Then FX (x) = GB0 (b0 (x; )) for all x 2 SX
, where b0 (:; ) is the strictly
increasing strategy in equilibrium that solves (2) with the boundary condition b0 (xL ; ) =
vh (xL ; xL ; ). Then the rationalizability of GB0 and the monotonicity of (:; GB0 ) implies
b0 (x; ) = 1 (vh (x; x; ); GB0 ) for all x 2 SX . It follows that (6) must hold. To prove
su¢ ciency, suppose GB0 is symmetric and a¢ liated with the support SBN
[b0L ; b0U ]N and
F that
(:; GB0 ) is increasing on the marginal support SB . Consider a = ( ; FX ) 2
satis…es (6). We need to show GB0 (b) = FX ( 0 (b; )) 8b 2 [b0L ; b0U ]N , where 0 (:; ) is the
inverse of the bidding strategy in a symmetric, increasing equilibrium b0 (:; ) that solves (1)
with the boundary condition b0 (xL ; ) = vh (xL ; xL ; ).38 The monotonicity of vh (x; x; ) in
38
Existence of symmetric, increasing PSBNE is not an issue since by the de…nition of
for all 2
F.
and F, they exist
45
implies GB0 (b) = FX (vh 1 ( (b; GB0 ); )) for all
x and (:; GB0 ) over SB and the choice of
b 2 [bL ; bU ]N .
Hence it su¢ ces to show 1 (vh (:; :; ); GB0 ) satis…es the …rst-order conditions in (1) with
the boundary condition 1 (vh (xL ; xL ; ); GB0 ) = vh (xL ; xL ; ). By the same argument as in
Proposition 1 in Li, Perrigne and Vuong (2002), it can be shown that limb!b0L (b; GB0 ) = b0L
under the rationalizable conditions on GB0 . Hence the boundary condition is satis…ed.
Let vh (x) be a shorthand for vh (x; x; ) and 1 () for 1 (:; GB0 ). Furthermore, from the
construction of ,
FY jX (xjx) = GM 0 jB 0 [
fY jX (xjx) = gM 0 jB 0 [
Thus
1
1
1
(vh (x))j
(vh (x))j
1
(vh (x))]
1
(vh (x))]
1
(vh (x))]
10
(vh (x))vh0 (x)
(vh (:)) solves (1) if
vh0 (x)
10
(vh (x)) = [vh (x)
fY jX (xjx)
FY jX (xjx)
or equivalently, if
1
vh (x) =
(vh (x)) +
GM 0 jB 0 [
gM 0 jB 0 [
1
(vh (x))j
1
(vh (x))j
But this must hold by the de…nition of (:; GB0 ).
1
1
(vh (x))]
(vh (x))]
Q.E.D.
N
Suppose (x) = (f~(xi ; yi )gN
maxj6=i xj . That is, bidders’
i=1 ) for all x 2 SX , where yi
valuations only depend on his own signal and the highest rival signal. Then vh (x; ; FX ) =
(f~(xi ; yi )gN
i=1 ) for all FX 2 F. Therefore, a distribution GB0 that satis…es conditions (i)
and (ii) is rationalized by any such 2
that satis…es the "maxj6=i Xj -su¢ ciency" with
0
~
boundary conditions (xk ; xk ) = (bk ) for k 2 fL; U g, and a signal distribution FX (x)
GB0 ( 1 (~(x1 ; x1 )); ::; 1 (~(xn ; xn ))).
Proof of Lemma 1. Note the solution to (3) with the boundary condition has the following
closed form:
Z b0 (x)
0
0
0
~
~ ~bjb0 (x))
rL(b (x (r))jb (x)) +
(~b; GB0 )dL(
r (b (x); GB0 )
b0 (x (r))
~ ~bjb; GB0 )
where L(
exp
Rb
~b
~ (u; GB0 )du . The proof of the lemma uses the monotonicity
and di¤erentiability of b0 (:). By change of variables,
fY jX (xjx)
FY jX (xjx)
= b00 (x) ~ (b0 (x); GB0 ) and for
46
~ 0 (s)jb0 (x); GB0 ). Furthermore, in equilibria vh (x; x; ; FX ) =
all s
x, L(sjx; FX ) = L(b
(b0 (x); GB0 ) for all x 2 SX . By de…nition, for x x (r),
Z x
r
vh (s; s; ; FX )L(sjx; FX ) (x; FX )dx
b (x) = rL(x (r)jx; FX ) +
x (r)
Z x
0
0
~ 0 (s)jb0 (x); GB0 ) ~ (b0 (s); GB0 )b00 (s)ds
~
(b0 (s); GB0 )L(b
= rL(b (x (r))jb (x); GB0 ) +
x (r)
=
r (b
0
(x); GB0 )
where the last equality follows from a change of variables in the integrand.
Proof of Lemma 2. The a¢ liation of signals and monotonicity of implies that vh (x; y) is
increasing in x and non-decreasing in y. For all x y,
Z y
Z y
fY jX (sjx)
fY jX (sjx)
vh (x; y)
vh (x; s)
vh (s; s)
ds v(x; y)
ds vl (x; y)
FY jX (yjx)
FY jX (yjx)
xL
xL
Therefore vh (xL ; xL ) = v(xL ; xL ) = vl (xL ; xL ) and v is bounded between vh and vl for all
x 2 SX . The proof of strict monotonicity of vh (x; x) in x is standard and omitted. For any
x < x0 on support, the law of total probability implies
vl (x0 ; x0 ) = E(vh (Y; Y )jXi = x0 ; Yi
= E(vh (Y; Y )jXi = x0 ; Yi
x0 )
x)P (Yi
E(vh (Y; Y )jXi = x0 ; x < Yi
xjXi = x0 ; Yi
x0 )P (x < Yi
x0 ) + :::
x0 jXi = x0 ; Yi
x0 )
By monotonicity of vh and x0 > x, E(vh (Y; Y )jXi = x0 ; x < Yi x0 ) > vl (x; x). By a¢ liation
of X and Y , E(vh (Y; Y )jXi = x0 ; Yi x) vl (x; x). Therefore vl (x0 ; x0 ) > vl (x; x).
Then (i) follows immediately. For the …rst part of (ii), note any rationalizable bid
distribution can also be rationalized by a certain private-value structure. It follows that 9 2
(GB0 ), vh (x; x; ) = v(x; x; ). For the second part of (ii), consider S f : (xi ; x i ) =
axi + (1
) maxj6=i xj for some a 2 (0; 1)g. By construction, all value functions in S are
exchangeable and non-decreasing in x i . Then vh (x; x) = x for all x 2 SX regardless of the
choice of signal distributions. Then de…ne FX (x) GB0 ( 1 (x1 ; GB0 ); ::; 1 (xn ; GB0 )). The
auction structure ( ; FX ) rationalizes GB0 in a Bayesian Nash equilibrium with the same FX
regardless of the choice of 2 (0; 1). By de…nition, v(x; x; ; FX ) = x + (1
)E(Yi jXi =
x; Yi
x) while vl (x; x; ; FX ) = E(Yi jXi = x; Yi
x). Therefore, the distance between v
and vl converges to 0 uniformly over the support SX as diminishes. It then follows that
8" > 0, 9 small enough such that supr2SRP jxh (r; ; FX ) x (r; ; FX )j " with FX de…ned
above.
47
Proof of Lemma 3. By the non-negativity of in , x (0; ) = xL for all 2
F.
0
0
Hence for all x xL , vh (x; x; ) = (b (x; ); GB0 ) and vl (x; x; ) = l (b (x; ); GB0 ) for all
0
2 (GB0 )
F. It follows (b; GB0 )
l (b; GB0 ) for all b 2 SB 0 , and (bL ; GB0 ) =
0
0
l (bL ; GB0 ). By the monotonicity of b and the de…nition of xl (r) and xh (r), b0 (x (r; ); ) 2
[b0 (xl (r; ); ); b0 (xh (r; ); )] for all
2
F. Furthermore, for all
2 (GB0 ) and
r 2 SRP ,
(b0 (xl (r; ); ); GB0 ) = vh (xl (r; ); xl (r; ); )
l (b
0
(xh (r; ); ); GB0 ) = vl (xh (r; ); xh (r; ); )
Note (; ; GB0 ) and l (; ; GB0 ) are invariant for all
2 (GB0 ), and l is increasing over
SB 0 by the monotonicity of vl (x; x) and b0 on SX . Therefore, b0k;r (GB0 ) = b0 (xl (r; ); ) for
k = l; h and all 2 (GB0 ), and Claim (i) in the lemma holds.
To prove Claim (ii), consider
f : (X) = Xi + (1
) maxj6=i Xj for some 2
(0; 1]g. Then vh (x; x; ) = x for all x 2 SX and 2 . Any rationalizable bid distribution
GB0 can be rationalized by the same signal distribution FX (x1 ; ::; xN ) GB0 ( 1 (x1 ; GB0 ); ::;
1
(xN ; GB0 )) for all 2 (0; 1]. Hence
fFX (GB0 )g
(GB0 ). By de…nition, x (r; ; FX ) =
2
arg minx2SX [r v(x; x; ; FX )] , where v(X; X; ) is continuous in x and , and x (r; ; FX )
is always a single-valued function in r. Hence the Theory of Maximum implies x (r; ; FX ) is
continuous in for all r 2 SRP and the FX chosen for GB0 . Furthermore for all 2
fFX g,
Rx
0
0
the equilibrium strategy b (x; ) = b (x; FX ) = xL sdL(sjx; FX ) is independent from .
Thus, enters the marginal bid b0 x r; ; FX only through the screening level, and the
marginal bid is also a continuous function in for all r 2 SRP and the chosen FX . Note
the image of a continuous mapping from any connected set is a connected set. Hence to
prove Claim (ii), it su¢ ces to show that the bounds are tight. Consider the case of a private
structure with = 1. Then
b0 (x (r; 1; FX ); FX ) = b0 (xl (r; 1; FX ); FX ) = b0l;r (GB0 )
and the lower bound is reached. On the other hand, the proof of part (ii) in Lemma 2 shows
xh (r; ; FX ) can be uniformly close to x (r; ; FX ) over r 2 SRP for some small enough.
As the choice of FX only depends on GB0 and is independent from , this suggests
supr2SRP jb0 (x (r; ; FX ); FX )
b0h;r (GB0 )j ! 0
as # 0. Combining these two results above shows for all r 2 SRP and b 2 [b0l;r (GB0 ); b0h;r (GB0 )),
9 2 (GB0 ) such that b0 (x (r; ); ) = b.
48
Proof of Lemma 4.
conditions is
r;k (b; GB0 )
Proof of (i): The closed form of solutions with new boundary
~ 0k;r (GB0 )jb; GB0 ) +
rL(b
for b
b0k;r (GB0 ). For any
[xl (r; ); xU ], and it follows
r
b (x; )
2
Z
b
~ ~bjb; GB0 )
(~b; GB0 )dL(
b0k;r (GB0 )
(GB0 ) and all x
rL(xl (r; )jx; FX ) +
Z
xl (r; ), vh (x; x; )
r 8x 2
x
vh (s; s; )dL(sjx; FX )
xl (r; )
Likewise, for all x
xh (r; ),
r
b (x; )
rL(xh (r; )jx; FX ) +
Z
x
vh (s; s; )dL(sjx; FX )
xh (r)
By non-negativity of , x (0; ) = xL and xh (r; ) x (r; ) xl (r; ) x (0; ) for all
r 2 SRP . Hence the equation (2) holds for xl (r; ) and xh (r; ). Substitution and the change
of variable show for all x xl (r; ),
~ 0l;r (GB0 )jb0 (x;
rL(b
r
b (x; )
and for all x
r
b (x; )
For all b
)) +
Z
b0 (x; )
~ ~bjb0 (x; ))
(~b; GB0 )dL(
r;l (b
0
(x; ); GB0 )
b0l;r (GB0 )
xh (r; ),
~ 0h;r (GB0 )jb0 (x; )) +
rL(b
Z
b0 (x; )
~ ~bjb0 (x; ))
(~b; GB0 )dL(
b0k;r (GB0 ) and k = l; h,
"
0
0
~
(b)
r;k (b; GB ) = (b)
~ 0 jb) +
rL(b
r;k
Z
b
Proof of (ii): Note for any t > r and
to the following minimization problem:
2
~ ~bjb)
(~b)dL(
b0r;k
It then follows from the monotonicity of the envelops f
(GB0 ), b0 ( r (t; ); ) 2 [ r;l1 (t; G0B ); r;h1 (t; G0B )].
0
r (br (
r;h (b
b0h;r (GB0 )
!#
0
(x; ); G0B )
>0
r;k (b; GB0 )gk=l;h
that for all
2
(GB0 ), b0 ( r (t; ); ) is de…ned as the solution
); t; GB0 ) = arg mins2[b0r (
);b0U ] [t
0
r (s; br (
); G0B )]2
where b0r ( ) is a shorthand for b0 (x (r; ) ; ) and r is de…ned as before. Fix the revenue
level t and the rationalizable bid distribution G0B . Then b0r ( ) can be treated as a parameter
49
that enters both the constraint set and the objective function, which is continuous in s and
b0r ( ). By construction, r;k1 (t; G0B ) = r (b0k;r (GB0 ); t; GB0 ). Furthermore, the Theorem of
Maximum implies r (b0r ( ); t; GB0 ) must be continuous in b0r ( ) for all t > r. Note by part
(ii) of Lemma 3, for all b 2 [b0l;r (GB0 ); b0h;r (GB0 )), 9 2 (GB0 ) such that b0r ( ) = b. It then
follows that for any r > r and for all b 2 [ r;l1 (t; G0B ); r;h1 (t; G0B )), 9 2 (GB0 ) such that
b0 ( r (t; ); ) = b.
Proof of Proposition 2. Recall from the lemmae above that b0 (x (r; ); ) 2 [b0l;r (GB0 );
b0h;r (GB0 )] for all 2 (GB0 ). By construction, r;k (b0k;r (G0B ); GB0 ) = br (x (r; ); ) = r.
Hence both f r;k (b0 (:; ); G0B )gk2fl;hg are invertible at t
r over the interval [xk (r; ); xU ]
1
1
0
0
for k 2 fl; hg and 2 (GB0 ). It follows that r;l (t; GB ) b0 (br 1 (t; ); )
r;h (t; GB ) for
t r and all 2 (GB0 ). The rest of the proof follows immediately.
Proof of Proposition 3.
First I prove that all
2
F and r 2 SRP , equilibrium
0
r
0
r
strategies b and b satisfy: (i) b (x; )
b (x; ) 8x
x (r; ) and (ii) the di¤erence
r
0
b (x; ) b (x; ) is decreasing in x for all x x (r; ). Then it follows immediately from
(i) and (ii) that FRI (r) ( ) F:S:D: F~RuI (r) (FRI (0) ( )). To prove (i), …rst note in equilibria
br (x (r; ); ) = r = v(x (r); x (r); )
E(Vi jXi = x (r); Yi
x (r))
b0 (x (r; ); ),
where the last inequality holds by equilibrium bidding conditions with no binding reserve
prices. Besides, for all x x (r; ), br (x; ) < b0 (x; ) implies b0r (x; ) > b00 (x; ). It follows
from Lemma 2 in Milgrom and Weber (1982) that br (x; ) b0 (x; ) for all x x (r; ). For
(ii), it su¢ ces to note sgn(b0r (x; ) b00 (x; )) = sgn(br (x; ) b0 (x; )) for all x x (r; ).
F jX (sjx)
To see that FRuI (r) (GB0 ) F:S:D: F~RuI (r) (FRI (0) ) in general, note FYY jX
F:S:D: L(sjx)
(xjx)
when private signals are a¢ liated. It follows that vl (x; x; ) b0 (x; ) for all x and therefore
0
xh (r; )
(r; ) for r b0L and 2 (GB0 ). Hence, b0 (xh (r); ) vl (xh (r); xh (r); ) = r
for r 2 [ 0L ; l (b0U )]. Furthermore, by a change-of-variables,
Z x
vh (s; s; )dL(sjx; )
r;h (b0 (x; ); GB0 ) = rL(xh (r)jx; ) +
xh (r)
and can be written as a solution for the di¤erential equation
0
(x) = [vh (x; x)
(x)]
fY jX (xjx)
FY jX (xjx)
with the boundary condition (xh (r)) = r for this range of r. An application of Lemma 2 in
Rx
Milgrom and Weber (1982) shows r;h (b0 (x; ); GB0 ) = rL(xh (r)jx)+ xh (r) vh (s; s)dL(sjx)
b0 (x) for all x xh (r; ). Hence r;h1 (t; GB0 ) t for t r and it follows FRuI (r) (GB0 ) F:S:D:
50
F~RuI (r) (FRI (0) ( )). For r > l (b0U ), the two upper bounds on the all-screening probability
coincide trivially at 1. To show that the bounds collapse into the same one with i.i.d.
signals, it su¢ ces to note that all inequalities in this proof so far will hold with equality as
F jX (sjx)
and L(sjx) are the same under i.i.d. signals.
the two distributions FYY jX
(xjx)
Proof of Proposition 4. Consider any
2
F. For notational ease, dependence of
r
x (r),
and vh on
is suppressed. By de…nition of v0 , PrfRII (r) < v0 g = 0. Note
PrfRII (r) = v0 g = PrfX (1) < x (r)g, Prfv0 < RII (r) < rg = 0 and PrfRII (r) = rg =
PrfX (1)
x (r) ^ X (2) < x (r)g. Since r0 (x) > 0 for all x 2 [x (r); xU ] and r (x (r)) =
vh (x (r); x (r)) r, it follows Prfr < RII (r) < vh (x (r); x (r))g = 0. Hence:
FRII (r) (t) = 0 8t < v0
= PrfX (1) < x (r)g 8t 2 [v0 ; r)
= PrfX (2) < x (r)g 8t 2 [r; vh (x (r); x (r)))
Next note for all t 2 [vh (x (r); x (r)); +1), PrfRII (r) 2 [vh (x (r); x (r)); t]g = Prfvh (X (2) ; X (2) ) 2
[vh (x (r); x (r)); t]g. Hence for all t in this range,
PrfRII (r)
PrfRII (r) < vh (x (r); x (r))g + PrfRII (r) 2 [vh (x (r); x (r)); t]g
tg =
= PrfX (2) < x (r)g + Prfvh (X (2) ; X (2) ) 2 [vh (x (r); x (r)); t]g
= Prfvh (X (2) ; X (2) )
tg
This characterizes the counterfactual distribution of RII (r). For any
Prfb0 (X (1) ; ) < b0l;r (GB0 )g
2
(GB0 ) and t < r,
FRl II (r) (t; GB0 )
PrfX (1) < x (r; )g = FRII (r) (t; )
Prfb0 (X (1) ; ) < b0h;r (GB0 )g
For r
FRuII (r) (t; GB0 )
t < vh (x (r); x (r)),
Prfvh (X (2) ; X (2) )
FRl II (r) (t; GB0 )
tg
Prfvh (X (2) ; X (2) ) < vh (x (r); x (r))g = FRII (r) (t; )
Prfb0 (X (2) ) < b0h;r (GB0 )g
FRuII (r) (t; GB0 )
0
since b0 (:) > 0. For t 2 [vh (x (r); x (r)); vh (xh (r); xh (r)),
FRl II (r) (t)
Prfvh (X (2) ; X (2) )
tg
FRII (r) (t; GB0 )
Prfvh (X (2) ; X (2) ) < vh (xh (r); xh (r))g = FRII (r) (t; )
= Prfb0 (X (2) ) < b0h;r (GB0 )g
FRuII (r) (t; GB0 )
51
due to the monotonicity of b0 (x) and vh (x; x) in x. For t 2 [vh (xh (r); xh (r)); +1),
FRl II (r) (t) = FRII (r) (t) = FRuII (r) (t) = Prfvh (X (2) ; X (2) )
tg
and this is because bids in 2nd-price auctions can be fully recovered from GB0 for those
who are not screened out under r. Finally, we complete the proof by noting 8 2 (GB0 ),
vh (x; x; ) = (b0 (x; ); GB0 ) for all x 2 [xL ; xU ].
Proof of Proposition 6. Auction characteristics are common knowledge among all bidders.
Hence in symmetric equilibria: (By symmetry among the bidders, bidder indices are dropped
for notational ease.)
@
b(x; z; n)
@X
= [~
vh (x; z; n)
f
(xjx;z;n)
b(x; z; n)] FYY jX;Z;N
jX;Z;N (xjx;z;n)
where v~h (x; z; n) E(Vi jXi = Yi = x; Z = z; N = n), Yi maxj6=i Xi , FY jX;Z;N (tjx; z; n)
Pr(maxj6=i Xj tjXi = x; Z = z; N = n) and fY jX;Z;N (tjx; z; n) is the corresponding conditional density. The equilibrium boundary condition for all (z; n) is b(xL ; z; n) = v~h (xL ; z; n).
For every z and n, the di¤erential equation is known to have the following closed form
solution :
Z x
b(x; z; n) =
h(z 0 ) + (s; n)dL(sjx; n)
xL
Independence of Xi and Z conditional on N implies both (x; n) and L(sjx; n) are invariant
to z for all s and x. Hence under assumptions A1’,A2 and A4,
b(xL ; z; n) = v~h (xL ; z; n) = h(z 0 ) + (xL ; n)
For x > xL , b(x; z; n) = h(z 0 ) +
Rx
(s; n)dL(sjx; n) for all (x; z; n).
xL
Proof of Proposition 7. Di¤erentiating br (x) for x
0
br (x; )
(x;FX )
For all r
0 and x; y
x (r) gives
(7)
+ br (x; ) = vh (x; x; )
x (r),
FY jX (yjx)
Pr(Y
= Pr(Y < x (r)jX = x) + Pr(x (r)
= Pr(br (Y ) < rjbr (X) = br (x)) + Pr(r
GrM jB (br (y)jbr (x))
(8)
yjx = x)
Y
yjX = x)
br (Y )
br (y)jbr (X) = br (x))
52
The equality of the two terms follows from the facts that Y < x (r) if and only if br (Y ) < r
and that br (x) is increasing for x
x (r). Taking derivative of both sides w.r.t. y for
y x (r) gives
r
(9)
fY jX (yjx) = b0r (y)gM
jB (br (y)jbr (x))
for all x; y
x (r). Substitute (9) and (8) into (7) proves the lemma.
Proof of Proposition 8. Let X (i:n) denote the ith largest signal among n potential bidders.
Then Pr(X (2:n) < x (r)jX (1:n)
x (r)) is observed. By the i.i.d. assumption, Pr(X (2:n) <
nFrn 1 (1 Fr )
x (r)jX (1:n)
x (r)) =
, where Fr
Pr(Xi
x (r)). The expression is
1 Frn
increasing in Fr . Therefore Fr is identi…ed, and Pr(X (1) < x (r)) = Frn .
9
Appendix B: Consistency of the three-step estimator
The lemma below extends the Basic Consistency Theorem of extreme estimators to those
de…ned over random, compact sets (as opposed to …xed, compact sets). The proof is an
adaptation from that of Theorem 4.1.1 in Amemiya (1985) and is included in Lemma A2 of
Li, Perrigne and Vuong (2003).
^ N (:) be nonstochastic and stochastic real-valued functions deLemma B1 Let Q(:) and Q
…ned respectively on compact intervals
[ l ; u ] and N [ lN ; uN ], where Prf[ lN ; uN ]
[ l ; u ]g = 1 for all N and kN ! k almost surely for k = l; u. For every N = 1; 2; :::; let
^N 2 N be such that Q
^ N (^N ) inf 2 Q
^ N ( ) + op (1). If Q(:) is continuous on
with a
N
p
l u
^ N ( ) Q( ) ! 0 as N ! +1,
unique maximizer on at 0 2 [ ; ] and (ii) sup 2 N Q
p
then ^N ! 0 .
9.1
Regularity properties of GM;B and gM;B
Let fY;X and FY;X denote the joint density and distribution of Yi and Xi respectively. Let
(:) be the bidding strategy in increasing, pure-strategy perfect Bayesian Nash equilibria.
Rx
Rx
That is, (x) = xL vh (s; s)dL(sjx) where L(sjx) = expf s FfYY XX(u;u)
dug. The lemma below
(u;u)
gives regularity results about the smoothness of the equilibrium bidding strategy.
53
Lemma B2 Under S1 and S2, (i) has R continuous bounded derivatives on [bL ; bU ] and
(:) c > 0 for some constant on [bL ; bU ]; (ii) GM;B and gM;B both have R 1 continuous
bounded partial derivatives on [bL ; bU ]2 .
0
Proof. First, I show that under S1 and S2, the equilibrium bidding function b0 (:) admits
up to R continuous, bounded derivatives on [xL ; xU ], and b00 (:) is bounded below from zero
f
(xjx)
with the
on SX . Recall b0 solves the di¤erential equation b00 (x) = [vh (x; x) b0 (x)] FYY jX
jX (xjx)
0
boundary condition b (xL ) = vh (xL ; xL ). Under S1, the joint density f has R continuous
bounded derivatives on [xL ; xU ]. By symmetry of FX ,
Rx Rx
::: xL (x; x; x3 ; ::; xn )f (x; x; x3 ; ::; xn )dx3 :::dxn
x
Rx Rx
vh (x; x) = L
::: xL f (x; x; x3 ; ::; xn )dx3 :::dxn
xL
Under S1, the denominator is 0 if and only if x = xL . Since by S2, the product (:; :; :)f (:; :; :)
also has R continuous, bounded derivatives, vh (x; x) has R + n 2 continuous, bounded
f
(xjx)
derivatives on (xL ; xU ]. Furthermore, it can be shown that FYY jX
also has R + n 2
jX (xjx)
continuous, bounded derivatives on any compact subsets of (xL ; xU ]. Therefore, b0 (:) has
R + n 1 continuous, bounded derivatives on any compact subsets of (xL ; xU ]. As for the
boundary point xL , the proof proceeds by applying Taylor expansions of f around the zero
f
vector in the de…nition of FYY jX
, L(sjx) and v(x; x), and then showing has R continuous,
jX
bounded derivatives at x = xL . It is a direct extension from proof of Lemma A2 in Li,
0
Perrigne and Vuong (2002) and excluded here for brevity. That b0 is bounded away from
zero on [xL ; xU ] follows from the same arguments as in Lemma A2 in Guerre, Perrigne and
Vuong (2000) and not repeated here.
Let gM;B denote the joint density of equilibrium bids B 0 and highest rival bid M 0 ,
Rm
and de…ne GM;B (m; b)
g
(~b; b)d~b. The relevant support is SB2 = [b0L ; b0U ]2 where
bL M;B
b0L = b0 (xL ) = vh (xL ; xL ) = (xL ) and b0U = b0 (xU ). (For notational ease, below I use
vh (x) as a shorthand for vh (x; x).) In equilibrium, b0 (vh 1 ( (b))) = b for all b 2 SB . Hence
0
0
(b) = fb00 [vh 1 ( (b))]vh 1 ( (b))g 1 where both vh 10 (:) and b00 (:) are bounded away from zero
and have R 1 continuous derivatives under S2. This proves part (i). For part (ii), note
Pr(M m; B b) = Pr(Y
vh 1 ( (m)); X vh 1 ( (b))) by the monotonicity of b0 (:). Hence
@
Pr(M
m; B
b) = vh 10 ( (b)) 0 (b) Pr(Y
vh 1 ( (m)); X = vh 1 ( (b))),
GM;B (m; b) = @B
2
where Pr(Y
y; X = x) has R + n 1 bounded, continuous derivatives on SX
and the …rst
2
two terms have R 1 continuous bounded derivatives on [bL ; bU ] as shown above. Besides,
@
gM;B (m; b) = @M
GM;B (m; b) = vh 10 ( (b)) 0 (b)vh 10 ( (m)) 0 (m)fY;X [vh 1 ( (m)); vh 1 ( (b))], where
the joint density fY;X has R + n 2 bounded, continuous derivatives on SB2 and vh 10 (:) has
R 1 continuous derivatives on SX . Hence gM;B (m; b) has R 1 continuous derivatives on
SB2 .
54
9.2
Consistency of ^b0l;r and ^b0h;r
^ M;B
The following lemma establishes the rate of uniform convergence of kernel estimates G
2
~ M;B to GM;B over S^2 . It is a preliminary step
and g^M;B to GM;B and gM;B over SB;
, and G
B;
for proving uniform convergence of ^l , ^ and ^l;r , ^h;r .
Lemma B3 Let hG = cG (log L=L)1=(2R+2n
S1-3,
^ M;B
supSB;
jG
2
5)
and hg = cg (log L=L)1=(2R+2n
1
GM;B j = O(hR
G );
~ M;B (b; b)
supb2S^B; jG
supSB;
j^
gM;B
2
1
GM;B (b; b)j = Op (hR
)
g
4)
. Under
1
gM;B j = O(hR
)
g
Furthermore, if R > n,
sup~bL
b;b2SB;
j
GM;B (~bL ; b)
j = Op (hR
g
GM;B (b; b)
^ M;B (~bL ; b)
G
^ M;B (b; b)
G
n
)
1
^ M;B GM;B = O(hR 1 ) and supS 2 j^
Proof.
That supSB;
G
gM;B gM;B j = O(hR
)
2
g
G
B;
follows from Lemma A5 in Li, Perrigne and Vuong (2002). By triangular inequality, for all b 2
Rb
^ M;B (~bL ; b) GM;B (~bL ; b) .
~ M;B (b; b) GM;B (b; b)
gM;B (t; b) gM;B (t; b)j dt+ G
S^B; , G
~bL j^
Thus
supb
jbU
~bL ;b2SB;
bL j supt
~ M;B (b; b)
jG
2
b;(t;b)2SB;
since by construction S^B;
Furthermore, note:
1f~bL
bg
inf b
j^
gM;B (t; b)
1
R 1
gM;B (t; b)j + Op (hR
)
G ) = Op (hg
SB; with probability 1 and hG < hg for L large enough.
^ M;B (~bL ;b)
G
^ M;B (b;b)
G
1f~bL bg
^
GM;B (b;b)
GM;B (b; b)j
GM;B (~bL ;b)
GM;B (b;b)
^ M;B (~bL ; b)
G
1f~bL bg
jG^ M;B (b;b)j
GM;B (~bL ; b) +
^ M;B (~bL ; b)
G
~
bL
GM;B (~bL ;b)
GM;B (b;b)
GM;B (b; b)
GM;B (~bL ; b) + GM;B (b; b)
^ M;B (b; b)
G
^ M;B (b; b)
G
Thus
sup~bL
b;b2SB;
j
^ M;B (~bL ;b)
G
^ M;B (b;b)
G
1
^ M;B (b;b)j
inf b2SB; jG
(
GM;B (~bL ;b)
j
GM;B (b;b)
sup~bL
b;b2C 2 (B)
sup~bL
^ M;B (~bL ; b)
jG
b;b2C (B)
GM;B (~bL ; b)j + :::
^ M;B (b; b)j
jGM;B (b; b) G
)
55
where
^ M;B (b; b)j
inf b2SB; jG
inf b2SB; jGM;B (b; b)j
^ M;B (b; b)
supb2SB; jG
GM;B (b; b)j
1
= inf b2SB; jGM;B (b; b)j + Op (hR
G )
Rb Rb
Note GM;B (b; b) = bL ::: bL g(b; b2 ; b3 ; ::; bn ) db2 :::dbn and g(b1 ; ::; bn ) = f (b0; 1 (b1 ); ::; b0; 1 (bn ))
has R continuous derivatives on [b0L ; b0U ]n . Since R > n, we can apply a Taylor expansion of
g(:) around (b0L ; :; b0L ) to get GM;B (b; b) = a(b bL )n 1 + o(jb bL jn 1 ) with a g(b0L ; ::; b0L ) >
n 1
0. It can then be shown inf b2SB; jGM;B (b; b)j
for some > 0 and = max(hg ; hG ).39
^ M;B (b; b)
Since R > n and = hg for L large enough, we have inf b2S
G
hn 1 +op (hn 1 ).
g
B;
Since sup~bL
b;b2SB;
^ M;B (~bL ; b)
jG
GM;B (~bL ; b)j and sup~bL
1
are both bounded by Op (hR
), it follows that sup~bL
g
g
~
^
~
GM;B (bL ; b)j
b;b2SB; jGM;B (bL ; b)
^ M;B (~bL ;b) GM;B (~bL ;b)
G
n
).
j G^ (b;b) GM;B (b;b) j = Op (hR
g
M;B
b;b2SB;
2
.
The next lemma proves the uniform convergence of ^ and ^l over the support SB;
Lemma B4 Let hG = cG (log L=L)1=(2R+2n 5) and hg = cg (log L=L)1=(2R+2n 4) . Under
R (n 1)
S1-3, supb2SB j^(b)
(b)j = Op (hg
) if R > n. Furthermore if R > 2(n 1),
R
2(n
1)
supb ~bL ;b2SB j^l (b)
).
l (b)j = Op (hg
Proof. Proposition A2 (ii) in Li, Perrigne and Vuong (2002) showed supb2C
R (n 1)
(b)j = Op (hg
). By de…nition, ~bL 2 SB; . Note :
sup~bL
2
b;(~bL ;b)2SB;
sup~bL
2
b;(~bL ;b)2SB;
sup~bL
j^l (b)
l (b)j
Z b
^(t) g^M;B (t;b) dt
~
G
(b;b)
2
b;(~bL ;b)2SB;
~bL
Z
M;B
^(~bL ) G^ M;B (~bL ;b)
^
G
(b;b)
M;B
Z
b
~bL
~bL
g
(B)
j^(b)
(t;b)
(t) GM;B
dt + :::
M;B (b;b)
g
(t;b)
(t) GM;B
dt
M;B (b;b)
bL
Below I show the two terms converge in probability to 0. By de…nition, ~bL
b0L , and
R ~bL
~
~
g
(t;b)
G
(bL ;b)
G
(bL ;b)
(t) GM;B
dt is bounded between (b0L ) GM;B
and (~bL ) GM;B
. With probability
b0L
M;B (b;b)
M;B (b;b)
M;B (b;b)
1,
sup~bL
2
b;(~bL ;b)2SB;
maxfsup~bL
39
^(~bL ) G^ M;B (~bL ;b)
^
G
(b;b)
2
b;(~bL ;b)2SB;
For details, see Lemma A6 in Li et.al 2002.
M;B
Z
~bL
bL
T1 (b; ~bL ); sup~bL
g
(t;b)
(t) GM;B
dt
M;B (b;b)
2
b;(~bL ;b)2SB;
T2 (b; ~bL )g
56
0 GM;B (~bL ;b)
^(~bL ) G^ M;B (~bL ;b)
~
(b
L ) GM;B (b;b) and T2 (b; bL )
^
GM;B (b;b)
2
such that b ~bL ,
With probability 1, for all (b; ~bL ) 2 SB;
^(~bL ) G^ M;B (~bL ;b)
^
G
(b;b)
where T1 (b; ~bL )
T1 (b; ~bL )
where
GM;B (~bL ;b)
GM;B (b;b)
sup~bL
^ M;B (~bL ;b)
G
^ M;B (b;b)
G
^(~bL )
GM;B (~bL ;b)
GM;B (b;b)
+
M;B
GM;B (~bL ;b)
GM;B (b;b)
^(~bL )
G
(~bL ;b)
(~bL ) GM;B
.
M;B (b;b)
(b0L )
1 by construction. Thus
T1 (b; ~bL )
2
b;(~bL ;b)2SB;
sup~bL
2
b;(~bL ;b)2SB;
p
Note ^(~bL ) ! j (b0L )j < 1 and ^(~bL )
8
<
GM;B (~bL ;b)
GM;B (b;b)
^ M;B (~bL ;b)
G
^ M;B (b;b)
G
:
^(~bL )
(b0L )
9
^(~bL ) + ::: =
;
p
(bL ) ! 0 by the uniform convergence of ^ over
^
p
p
G
(~bL ;b)
GM;B (~bL ;b)
SB; and that ~bL ! b0L . By Lemma A3, sup~bL b;(~bL ;b)2SB; G^M;B (b;b)
!
0 if
G
(b;b)
M;B
M;B
p
p
R > n. Hence sup~bL b;(~bL ;b)2S 2 T1 (b; ~bL ) ! 0 if R > n. That sup~bL b;(~bL ;b)2S 2 T2 (b; ~bL ) ! 0
B;
B;
follows from similar arguments.
2
By the triangular inequality, for all b ~bL , and (~bL ; b) 2 SB;
Z b
Z b
gM;B (t; b)
^(t) g^M;B (t; b) dt
(t)
dt
~ M;B (b; b)
GM;B (b; b)
~bL
~bL
G
9
8 R
=
< ~b ^(t)^
+
:::
g
(t;
b)dt
(t)g
(t;
b)dt
M;B
M;B
bL
1
~ M;B (b;b)j
inf ~b b;(~b ;b)2S 2 jG
~ M;B (b; b) GM;B (b; b)
;
:
(b0 ) G
L
L
B;
U
Rb
g
(t;b)
~ M;B (b; b)
where I use ~bL (t) GM;B
jGM;B (b; b)j
dt
(b0U ). Also note G
M;B (b;b)
hng 1 + op (hng
It is shown in Lemma B3 above that inf b2SB; jGM;B (b; b)j
~ M;B (b; b) GM;B (b; b)j = Op (hR 1 ). Furthermore for
and sup~bL b;(~bL ;b)2S 2 jG
g
B;
(~bL ; b) 2 S 2 ,
~ M;B (b; b) GM;B (b; b) .
G
1
) with R > n
all b ~bL , and
B;
Z
Z
b
~bL
b
^(t)^
gM;B (t; b)dt
^(t)
(t)gM;B (t; b)dt
(t) j^
gM;B (t; b)j dt +
~bL
Z
b
~bL
The boundedness of gM;B and implies
Z b
^(t)^
sup~bL b;(~bL ;b)2S 2
gM;B (t; b)dt
B;
which is the rate of convergence of supb2SB; ^
sup~bL
2
b;(~bL ;b)2SB;
j
b
~bL
^(t) g^M;B (t;b) dt
~
G
(b;b)
M;B
gM;B (t; b)j dt
(t)gM;B (t; b)dt = Op (hR
g
~bL
Z
j (t)j j^
gM;B (t; b)
(n 1)
)
. As a result,
Z
b
~bL
g
(t;b)
(t) GM;B
dtj = Op (hR
g
M;B (b;b)
2(n 1)
)
57
and converges to zero when R > 2(n
1).
The proposition below establishes the consistency of ^b0k;r using Lemma B1.
Proposition B1 Let hG = cG (log L=L)1=(2R+2n 5) and hg = cg (log L=L)1=(2R+2n 4) . Under
p
p
S1-3, ^b0l;r ! b0l;r (GB0 ) if R > n 1 and ^b0h;r ! b0h;r (GB0 ) if R > 2(n 1) for all r 2 SRP .
! 0. Hence
Proof. First note that ~bk ! b0k almost surely for k = L; U , and that
a:s:
^bL ! b0 . It su¢ ces to show that for all r 2 SRP , (i) (^(b) r)2 and (^ (b) r)2 converge
l
L
2
2
2
^
in probability to ( (b) r) and ( l (b) r) uniformly over SB; ; and (ii) ( (b) r)2 and
( l (b) r)2 are continuous on [b0L ; b0U ] with unique minimizers b0l;r and b0h;r respectively on
p
p
[b0L ; b0U ]. By Lemma B4, supb2SB; j^(b) (b)j ! 0 and supb ~bL ;(~bL ;b)2S 2 j^l (b) l (b)j ! 0.
B;
And
supb2SB; (^(b)
r)2
( (b)
r)2
2
supb2SB; ^ (b)
2
(b) + 2r supb2SB; ^(b)
(b)
where both terms converge to 0 in probability since supb2SB; (b)
(b0U ) < 1. Likewise
p
supb ~bL ;(~bL ;b)2S 2 (^l (b) r)2 ( l (b) r)2 ! 0 by similar arguments. Next, the continuB;
ity of ( (b) r)2 and ( l (b) r)2 follows from the smoothness of . Also both (:; GB0 ) and
0
0
l (:; GB0 ) are increasing on [bL ; bU ] by the monotonicity of vh (:; :; ) and vl (:; :; ) as well as
b0 (:; ) on SX for all 2 (GB0 ). Thus for all r 2 SRP , the minimizers of ( (b) r)2 and
( l (b) r)2 are unique on [b0L ; b0U ].
9.3
Uniform convergence of ^k;r (:; ^b0k;r )
Lemma B5 Let hG = cG (log L=L)1=(2R+2n
and if R > 2n 1,
g^M;B (b; b)
^ M;B (b; b)
G
supb2SB;
5)
and hg = cg (log L=L)1=(2R+2n
gM;B (b; b)
= Op (hR
GM;B (b; b)
2n+1
4)
. Under S1-3
)
2
Proof. The proof is similar to the last part of Lemma B3. On the support SB;
,
g^
j G^M;B
M;B
gM;B
j
GM;B
1
jG^ M;B jjGM;B j
jGM;B j j^
gM;B
^ M;B
gM;B j + jgM;B j G
GM;B
1
^ M;B GM;B j = Op (hR 1 ) and supS 2 j^
By Lemma B3, supSB;
jG
gM;B gM;B j = Op (hR
).
2
g
G
B;
Besides, supSB;
jGM;B j < 1 and supSB;
jgM;B j < 1 implies the supremum of the term in the
2
2
g^
(b;b)
1
bracket is Op (hR
) as hg > hG for L large enough. Hence supb2SB; j G^M;B (b;b)
g
M;B
gM;B (b;b)
j
GM;B (b;b)
58
1
)
Op (hR
g
, where the two terms in the denominator
^
inf b2SB; jGM;B j inf b2SB; jGM;B j
hng 1 + o(hng 1 ) and hng 1 + o(hng 1 ) respectively by some constant
2
2
denominator is bounded below by h2n
+ o(h2n
). Hence supb2SB;
g
g
Op (hR 2n+1 ) and converges in probability to 0 if R > 2n 1.
are bounded below by
and . It follows the
gM;B (b;b)
j=
GM;B (b;b)
g^
(b;b)
j G^M;B (b;b)
M;B
Lemma B6 Let hG = cG (log L=L)1=(2R+2n 5) and hg = cg (log L=L)1=(2R+2n 4) . Under S1p
0
! 0 for all r 2 SRP .
3 and suppose R > 2n 1, then supb2SB; ^k;r (b; ^b0k;r )
k;r (b; bk;r )
Proof. Consider the case where r is in the interior of SRP (or equivalently, b0k;r 2 (b0L ; b0U )).
By de…nition, ^b0k;r 2 SB; , and for sample sizes large enough and small enough, b0k;r is in
the interior of SB; . By the triangular inequality,
supb2SB; ^k;r (b; ^b0k;r )
supb2SB; 1(^b0k;r
0
k;r (b; bk;r )
b0k;r )1(b > b0k;r ) ^k;r (b)
k;r (b)
+ :::
supb2SB; 1(^b0k;r > b0k;r )1(b > ^b0k;r ) ^k;r (b)
k;r (b)
+ :::
supb2SB; 1(^b0k;r
b0k;r )1(b 2 (^b0k;r ; b0k;r ]) ^k;r (b)
supb2SB; 1(^b0k;r > b0k;r )1(b 2 (b0k;r ; ^b0k;r ]) j
k;r (b)
r + :::
rj
It su¢ ces to show all four terms (denoted A1 , A2 , A3 and A4 respectively) converge in
probability to 0 uniformly over b 2 SB; as sample size increases. For A1 ,
supb2SB; 1(^b0k;r
supb0k;r
b b0U
0
b0k;r )1(b > b0k;r ) ^k;r (b; ^b0k;r )
k;r (b; bk;r )
8
Rb
^(t) ^ (t)L(tjb)
^
>
(t) (t)L(tjb)dt + ::
<
b0k;r
0
0
^
1(bk;r bk;r )
R b0
>
^ ^b0 jb) L(b0 jb)
^
+ r L(
: ^b0k;r ^(t) ^ (t)L(tjb)dt
k;r
k;r
k;r
p
9
>
=
>
;
^
It can be shown supt b;(t;b)2SB;
L(tjb)
L(tjb) ! 0 using convergence results from previ2
ous lemmae.40 Note for r in the interior of SRP ,
supb0k;r <b
sup
b0k;r <b
1(^b0k;r
sup
^ ^b0 jb)
b0k;r )r L(
k;r
L(b0k;r jb)
1(^b0k;r
^ ^b0k;r jb)
b0k;r )r L(
L(^b0k;r jb) + :::
1(^b0k;r
b0k;r )r L(^b0k;r jb)
L(b0k;r jb)
bU
b0k;r <b bU
40
bU
For details of the proof, see Li, Perrigne and Vuong (2003).
59
For su¢ ciently small , b0k;r > b0L + . By construction ^b0k;r 2 SB; , and therefore supb0k;r <b
^ ^b0 jb)
b0k;r )r L(
k;r
bU
1(^b0k;r
p
L(^b0k;r jb)
! 0. Also by the mean value theorem,
L(^b0k;r jb)
L(b0k;r jb)
@
L(tjb)jt=~b0 (^b0k;r b0k;r )
k;r
@t
(~b0k;r )L(~b0k;r jb) ^b0k;r b0k;r
=
=
for some ~b0k;r between ^b0k;r and b0k;r . The consistency of ^b0k;r for b0k;r suggests ~b0k;r is bounded
away from b0L as sample size increases. Thus both (~b0k;r ) and L(~b0k;r jb) converge in probability to some …nite constant since supSB; jgj < 1 and inf SB; jg 0 j > c > 0 and hence
supb0 b b
1(^b0
b0 ) L(^b0 jb) L(b0 jb) is op (1). Next
U
k;r
k;r
k;r
supb0k;r <b
b0U
k;r
Z
bU
k;r
b
^(t) ^ (t)L(tjb)
^
(t) (t)L(tjb)dt
b0k;r
b0k;r supb0k;r
^(t) ^ (t)L(tjb)
^
t b b0U
(t) (t)L(tjb)
p
!0
^ over SB; and
where the right hand side is op (1) by the uniform convergence of ^, ^ , and L
2
SB;
under S1-3 and the boundedness of ,
and L on the closed interval [b0k;r ; b0U
].
Finally
sup
b0k;r <b
1(^b0k;r
b0k;r )
bU
sup
2
t b;(t;b)2SB;
Z
b0k;r
^(t) ^ (t)L(tjb)dt
^
^b0
k;r
^(t) ^ (t)L(tjb)
^
1(^b0k;r
b0k;r ) ^b0k;r
b0k;r
p
! 0, and supb0k;r <b b0U ;t b;(t;b)2SB;
1(^b0k;r
2
p
t b0k;r ) j (t) (t)L(tjb)j is bounded with probability approaching 1 as ^b0k;r ! b0k;r . Therefore
R b0k;r
^(t) ^ (t)L(tjb)dt
^
supb0k;r <b bU
= op (1) and it follows A1 = op (1). For A2 ,
^b0
Note supt
2
b;(t;b)2SB;
^(t) ^ (t)L(tjb)
^
(t) (t)L(tjb)
k;r
supb2SB; 1(^b0k;r > b0k;r )1(b > ^b0k;r ) ^k;r (b)
k;r (b)
8
Rb
^(t) ^ (t)L(tjb)
^
>
(t) (t)L(tjb)dt + :::
<
^b0
k;r
0
0
^
sup
1(b > bk;r > bk;r )
R ^b0
>
^ ^b0 jb) L(b0 jb)
b0k;r <b b0U
: b0k;r (t) (t)L(tjb)dt + r L(
k;r
k;r
k;r
where the supremum of the …rst and last term over b0k;r < b
b0U
9
>
=
>
;
are op (1) by the same
60
argument as above, and
supb0k;r
b bU
supb0k;r
b bU
1(b > ^b0k;r > b0k;r )
2
;(t;b)2SB;
Z
b
^(t) ^ (t)L(tjb)
^
(t) (t)L(tjb)dt
^b0
k;r
^(t) ^ (t)L(tjb)
^
(t) (t)L(tjb) b
^b0 = op (1)
k;r
For A3 ,
supb2SB; 1(^b0k;r
= supb2SB;
1(^b0k;r
b0k;r )1(b 2 (^b0k;r ; b0k;r ]) ^k;r (b)
b0k;r )1(b
r
^ ^b0 jb)
2 (^b0k;r ; b0k;r ]) rL(
k;r
r+
Z
b
^(t) ^ (t)L(tjb)dt
^
^b0
k;r
2
^
By construction, ^b0k;r 2 SB; . The uniform convergence of L(tjb)
for all t
b on SB;
, the
p
0
0
0
0
^
^
continuity of L(tjb) in both arguments, and bk;r ! bk;r suggest supb2SB; 1(bk;r bk;r \ b 2
^ ^b0 jb) r = op (1). Also for large samples, ^b0 is bounded away from b0 with
(^b0k;r ; b0k;r ]) rL(
L
k;r
k;r
R
b
^
probability approaching to 1 and supb2SB; 1(^b0k;r b0k;r \b 2 (^b0k;r ; b0k;r ]) ^b0 ^(t) ^ (t)L(tjb)dt
k;r
= op (1). For A4 , note k;r is continuous at b0k;r with k;r (b0k;r ) = r and is increasing beyond
p
b0k;r . Hence the consistency of ^b0k;r is su¢ cient for A4 ! 0. In the boundary case where
b0k;r = b0L , it su¢ ces to show the convergence of terms A2 and A4. The same argument above
applies.
9.4
Final step of the proof
Lemma B7 Let hG = cG (log L=L)1=(2R+2n 5) and hg = cg (log L=L)1=(2R+2n 4) . Under S1-3
1
p
and suppose R > 2n 1, for any r 2 Sr , ^k;r (t; ^b0k;r ) ! k;r1 (t; b0k;r ) for k = fl; hg and all
t > r.
Proof. For the range of r 2 SRP and t > r,
1
0
k;r (t; bk;r )
are unique minimizers of [
k;r (b)
t]2
p
0
over SB . Lemma B6 showed that supb2SB; ^k;r (b; ^b0k;r )
! 0 and k;r (:; b0k;r )
k;r (b; bk;r )
is also continuous on SB . Also ~bk ! b0k almost surely for k = L; U as sample size increases.
All conditions for Lemma B1 are satis…ed and claim is proven.
P
Lemma B8 Let F^n (t) = n1 ni=1 1(Zi
t) where fZi gni=1 is an i.i.d. sample from a
a:s
population distributed as FZ . Then supt2R jF^n (t) FZ (t)j ! 0. If FZ (t0 ) is continuous
p
at t0 and a sequence of random variable t^n ! t0 and t0 is a continuity point of F , then
p
F^n (t^n ) ! FZ (t0 ).
61
Proof. The …rst claim follows from Glivenko-Cantelli Lemma and the proof of the second
claim is standard (e.g. see Theorem 4.1.5 Amemiya 1985).
The proof of Proposition 4 follows directly from results of the lemmae above.
Proof of Proposition 5.
1
Ln
probability to Pr(Blmax b) uniformly
t in the stated range of interests. The
p
u
u
and F^R(r)
(t) ! FR(r)
(t) for given r and t.
10
PLn
max
b)converge in
l=1 1(Bl
1
p
over SB . By Lemma B7, ^r;k (t) ! r;k1 (t) for all r and
p
l
l
second part of Lemma B8 proves F^R(r)
(t) ! FR(r)
(t)
By the …rst part of Lemma B8,
Appendix D: Derivations for Monte Carlo designs
A. Closed forms of vl (x; x; c) and b0 (x; c) in Design 1
Let Xi = X0 + "i for i = 1; 2, where "i are statistically independent from (X0 ; " i ). Let
"i be uniform on [ c; c] and X0 be uniform on [a; b]. For the closed form of vl and b0 , we
need to calculate the conditional expectation vl (t; t) = E(X2 jX2 t; X1 = t) and the inverse
f
(t)
1 =t
hazard ratio FXX2 jX
. For any t1 ; t2 such that t2 2 [t1 2c; t1 + 2c],
(t)
jX
=t
2 1
Z
fX2 jX1 =t1 (t2 ) =
fX2 jX0 =s;X1 =t1 (t2 )fX0 jX1 =t1 (s)ds
(10)
S(t1 )
Z
=
f"2 (t2 s)fX0 jX1 =t1 (s)ds
S(t1 )
where S(t1 ) = [max(a; t1 c); min(b; t1 + c)] is the support of X0 given X1 = t1 . The second
equality follows from the independence of "2 from (X0 ; "1 ). The conditional density of X0
given X1 = t1 depends on values of a, b and c. In the case b a 2c,
fX0 jX1 =t1 ~
U nif [a; t1 + c]
fX0 jX1 =t1 ~
U nif [t1
c; t1 + c]
fX0 jX1 =t1 ~
U nif [t1
c; b]
and for the case b
8 t1 2 [a
c; a + c]
8 t1 2 [a + c; b
8 t1 2 [b
c]
c; b + c]
a < 2c,
fX0 jX1 =t1 ~
U nif [a; t1 + c]
fX0 jX1 =t1 ~
U nif [a; b]
fX0 jX1 =t1 ~
U nif [t1
8 t1 2 [a
8 t1 2 [b
c; b]
c; b
c]
c; a + c]
8 t1 2 [a + c; b + c]
62
Therefore equation (10) suggests conditional on X1 = t1 , X2 is distributed as the sum of two
independent uniform variables with support on S(t1 ) and [ c; c] respectively.
First consider the case b
2c. If t1 2 [a
a
1
2c
1
=
2c
1
=
2c
fX2 jX1 =t1 (t2 ) =
If t1 2 [a + c; b
t2
t1
c; a + c),
a+c
a+c
8t2 2 [a
8t2 2 [t1 ; a + c]
t1 + 2c
t1 + c
t2
a
8t2 2 [a + c; t1 + 2c]
c],
1
(t2 t1 + 2c)
4c2
1
=
(t1 + 2c t2 )
4c2
8t2 2 [t1
fX2 jX1 =t1 (t2 ) =
If t1 2 (b
2c; t1 ]
8t2 2 [t1 ; t1 + 2c]
c; b + c],
1
2c
1
=
2c
1
=
2c
fX2 jX1 =t1 (t2 ) =
Next consider the case b
t2 t1 + 2c
b + c t1
8t2 2 [t1
2c; b
8t2 2 [b
b+c
b+c
a < 2c. If t1 2 [a
1
2c
1
=
2c
1
=
2c
t2
t1
1
2c
1
=
2c
1
=
2c
t2
fX2 jX1 =t1 (t2 ) =
If t1 2 [b
c; t1 ]
t2
t1
c; b
a+c
a+c
c]
c; t1 ]
8t2 2 [t1 ; b + c]
c],
8t2 2 [a
c; t1 ]
8t2 2 [t1 ; a + c]
t1 + 2c
t1 + c
t2
a
8t2 2 [a + c; t1 + 2c]
c; a + c],
fX2 jX1 =t1 (t2 ) =
a+c
b a
8t2 2 [a
8t2 2 [b
b + c t2
b a
c; b
c]
c; a + c]
8t2 2 [a + c; b + c]
63
If t1 2 [a + c; b + c],
1
2c
1
=
2c
1
=
2c
t2 t1 + 2c
b + c t1
fX2 jX1 =t1 (t2 ) =
8t2 2 [t1
2c; b
8t2 2 [b
b+c
b+c
t2
t1
c]
c; t1 ]
8t2 2 [t1 ; b + c]
1
8t1 2 [a c; b + c] for both cases. Furthermore FX2 jX1 =t1 (t1 )
To sum up, fX2 jX1 =t1 (t1 ) = 2c
can be calculated by integrating the corresponding densities over proper ranges. That is,
Rt
FX2 jX1 =t1 (t1 ) = 0 1 fX2 jX1 =t1 (s)ds. Thus both screening level x (r; c) can be calculated and
equilibrium bids b0 (x (r; c); c) can be calculated using numerical approximations.
B. Closed form of vl (x; x) in Design 2
Distribution and density of truncated normal distributions (with the underlying normal
distribution being N ( ; ) are respectively:
~ (x) =
(
(
x
)
xU
(
)
xL
(
1
)
~ (x) =
;
xL
)
(
xU
(
x
)
)
(
xL
)
where and denote the density and distribution of the parental (untruncated) normal distribution with mean and standard deviation , and (xL ; xU ) denotes the pair of truncation
points. Suppose X is distributed as truncated normal on (xL ; xU ), then for all a 2 (xL ; xU ),
Ra x x
Ra
~ (x)dx
(
)dx
x
xL
xL
E(XjX a)
=
a
xL
~ (a)
(
)
(
)
Note
@
x
(
@X
)=
(
=
=) (
Then the numerator is
Z
x
a
(
)
x
x
(
=
(
) dx
( (
Z
a
a
)
(
xL
x
)(
x
x
xL
xL
=
x
)
)
x1
1
+ (
x
)
@
x
(
@X
@
x
(
@X
)dx
))
( (
a
)
1
)
)
(
xL
))
64
Therefore
( (
E(XjX
a
)
(
xL
a)
(
(
=
(
a
a
a
xL
)
(
)
(
xL
))
( (
)
(
)
)
xL
a
)
)
(
xL
))
65
References
[1] Athey, S. and P. Haile (2002), "Identi…cation of Standard Auction Models," Econometrica 70:2107-2140
[2] Athey, S. and P. Haile (2005), "Nonparametric Approaches to Auctions," Handbook of
Econometrics, Vol.6, forthcoming
[3] Bajari, P. and A. Hortacsu (2003), "The Winner’s Curse, Reserve Prices, and Endogenous Entry" Empirical Insights from eBay Auctions," RAND Journal of Economics
34:329-355
[4] Butler, Alexander W. (2007), "Distance Still Matters: Evidence from Municipal Bond
Underwriting"
[5] Guerre, E., I. Perrigne and Q. Vuong (2000), "Optimal Nonparametric Estimation of
First-Price Auctions," Econometrica 68:525-574
[6] Haile, P., H. Hong and M. Shum (2004) "Nonparametric Tests for Common Values in
First-Price Sealed Bid Auctions," mimeo, Yale University
[7] Haile, P. and E. Tamer (2003), "Inference with an Incomplete Model of English Auctions," Journal of Political Economy, vol.111, no.1
[8] Hardle, W. (1991), Smoothing Techniques with Implementation in S. New York :
Springer-Verlag, 1991
[9] Hendricks, K., J. Pinkse and R. Porter (2003), "Empirical Implications of Equilibrium
Bidding in First-Price, Symmetric, Common Value Auctions," Review of Economic
Studies 70:115-145
[10] Hendricks, K. and R. Porter (2007), "An Empirical Perspective on Auctions," Handbook
of Industrial Organization, Vol 3, Elsevier
[11] Hong, H and M. Shum (2002), "Increasing Competition and the Winner’s Curse: Evidence from Procurement"
[12] Ichimura, H. (1991), "Semiparametric Least Squares Estimation of Single Index Models," Journal of Econometrics
[13] Imbens and Manski (2004), "Con…dence Intervals for Partially Identi…ed Parameters",
Econometrica, Vol 72, No 6.
66
[14] Klemperer, P. (1999) "Auction Theory: A Guide to the Literature," Journal of Economic
Surveys 12:227-286
[15] Krishna, Vijay (2002), "Auction Theory", Academic Press
[16] La¤ont, J. and Q. Vuong (1996), "Structural Analysis of Auction Data," American
Economic Review, Papers and Proceedings 86:414-420
[17] Levin, D. and J. Smith (1994), "Optimal Reservation Prices in Auctions," Economic
Journal 106:1271-1283
[18] Li, T., I. Perrigne, and Q. Vuong (2002), "Conditionally Independent Private Information in OCS Wildcat Auctions", Journal of Econometrics 98:129-161
[19] Li, T., I. Perrigne, and Q. Vuong (2003), "Estimation of Optimal Reservation Prices in
First-price Auctions", Review of Economics and Business Statistics
[20] McAdams, D. (2006), "Uniqueness in First Price Auctions with A¢ liation," Working
Paper
[21] McAfee, R and J. McMillan (1987), "Auctions and Bidding," Journal of Economic
Literature 25:669-738
[22] Milgrom, P. and R. Weber (2000), "A Theory of Auctions and Competitive Bidding,"
Econometrica, 50:1089-1122
[23] Paarsch, H. (1997), "Deriving an Estimate of the Optimal Reserve Price: An Application
to British Columbian Timber Sales," Journal of Econometrics 78:333-57
[24] Powell, J., J. Stock and T. Stocker (1989) "Semiparametric Estimation of Index Coef…cients,”Econometrica, 57: 1403-1430
[25] Securities Industry & Financial Markets Association Publications, 2005
[26] Shneyerov, A. (2006) "An Empirical Study of Auction Revenue Rankings: the Case of
Municipal Bonds," Rand Journal of Economics, Forthcoming
[27] Temel, J. (2001), The Fundamentals of Municipal Bonds, 5th Edition, Wiley Finance
Appendix C : Graphs, Figures and Tables
Figure 1 (a)
Figure 1 (b)
Figure 1 (c)
Figure 1 (e)
Figure 1 (d)
Figure 1 (f)
Figure 2
Figure 3(a)
Figure 3(b)
Figure 3(c)
Figure 4(a)
Figure 4(b)
Figure 5(a)
Figure 5(b)
Figure 6
Figure 7
Figure 8(b)
Figure 9(a)
Figure 9(b)
Figure 9 (c)
Figure 9(d)
Table 1(a) : Descriptive Statistics
# of bidding syndicates
2
3
4
5
6
7
8
9
10
11
# of auctions
608
971
1075
1007
852
687
531
406
258
158
Average par value ($mil)
4.90
9.22
17.33
23.13
27.02
27.27
30.20
32.68
26.54
33.57
Average price (/ $100 par)
99.17
99.31
99.51
99.97 100.32 100.44 101.00 101.06 101.19 101.64
Average spread
0.87
1.15
1.18
1.20
1.18
Average bid
98.73
98.77
98.95
99.42
99.79
1.20
1.07
1.04
1.00
1.01
99.94 100.53 100.59 100.76 101.22
Std. dev.
1.67
1.84
1.95
2.54
2.58
2.98
2.97
2.90
3.07
2.97
Minimal bid
90.28
85.20
86.47
87.03
86.68
13.26
93.44
91.67
93.94
93.87
Maximal bid
113.93 110.59 108.85 119.16 111.66 114.55 111.45 111.87 128.00 111.45
Table 1 (b) : Descriptive Statistics
Prices
Min
Percentiles
1
10
20
30
40
50
60
70
80
90
99
Max
96.74
97.98
98.49
98.91
99.29
99.66
100.00
100.33
100.91
102.84
109.06
128.00
All Bids
Min
Percentiles
1
10
20
30
40
50
60
70
80
90
99
Max
95.32
97.37
98.07
98.56
98.99
99.40
99.81
100.28
101.14
103.54
109.30
128.00
# of auctions
6,721
# of bids
37,547
WA Coupon Rate
Min
1
10
20
30
40
50
60
70
80
90
99
Max
0.0100
0.0214
0.0322
0.0355
0.0377
0.0392
0.0405
0.0419
0.0435
0.0454
0.0481
0.0549
0.0671
Total Par Value
Min
1
10
20
30
40
50
60
70
80
90
99
Max
(in $million)
0.105
0.385
1.275
2.160
3.200
4.485
6.000
8.581
12.000
20.415
45.000
297.831
809.470
91.17
85.20
# of bidders
1
2
3
4
5
6
7
8
9
10
11
12+
Total
# of auctions
19
608
971
1075
1007
852
687
531
406
258
158
149
6721
0.28
9.05
14.45
15.99
14.98
12.68
10.22
7.90
6.04
3.84
2.35
2.22
100.00
SecType
Unlimited GO
Limited GO
Revenue
4334
1061
1326
64.484
15.786
19.729
Table 2 : Pooled Random Effect Estimates
Est
Std Err
t-stat
p-value
wacr
1.520
0.122
12.49
0.00
wapn
-1.037
0.061
-17.11
0.00
sectype
2.476
0.458
5.41
0.00
BQ
-0.837
0.056
-15.00
0.00
totpar
1.764
0.108
16.32
0.00
type_cr
-0.680
0.121
-5.64
0.00
HR
0.221
0.175
1.26
0.21
HR_pn
-0.002
0.067
-0.03
0.98
NE
0.428
0.142
3.00
0.00
SW
-0.188
0.188
-1.00
0.32
South
0.226
0.121
1.87
0.06
West
-0.323
0.149
-2.17
0.03
NE_rating
0.017
0.177
0.10
0.92
SW_rating
-0.038
0.231
-0.16
0.87
South_rating
0.309
0.175
1.77
0.08
West_rating
0.389
0.215
1.81
0.07
d3
94.920
0.450
210.97
0.00
d4
94.968
0.445
213.26
0.00
d5
95.323
0.438
217.61
0.00
d6
95.579
0.443
215.91
0.00
d7
95.738
0.438
218.44
0.00
d8
96.128
0.443
216.92
0.00
Number of cluster
F( 21, 5122)
5123.00
86.66
Prob > F
0.00
R-squared
0.43
Root MSE
1.98
Table 3(a) : GLS estimates for fixed number of potential bidders
number of bidders
4
5
6
7
8
Intercept
98.221
169.540
95.524
91.620
92.578
76.000
93.563
79.570
95.500
64.370
WA coupon rate
0.611
3.790
1.495
5.020
2.205
6.970
2.021
6.280
1.797
4.710
WA maturity
-0.896
-8.690
-1.082
-7.740
-1.025
-8.430
-0.944
-5.090
-1.278
-5.730
Security Type
0.264
0.390
3.297
3.110
4.514
3.960
3.945
3.370
2.180
1.400
Bank Qualified
-0.529
-5.510
-0.723
-6.580
-0.694
-5.330
-0.848
-5.820
-1.277
-6.110
Ratings
0.061
0.170
-0.115
-0.310
0.732
1.750
0.435
1.010
0.174
0.320
Type*WACR
-0.051
-0.290
-0.935
-3.330
-1.249
-4.250
-1.036
-3.380
-0.571
-1.430
Ratings*WAPN
0.113
0.900
0.219
1.500
-0.158
-1.210
-0.311
-1.530
0.080
0.330
Par amount
1.423
6.220
1.770
8.390
1.449
8.750
2.200
8.570
2.182
6.430
N.E.
0.065
0.280
0.836
2.810
0.905
2.670
0.149
0.280
0.380
0.860
South
-0.225
-1.130
-0.052
-0.220
0.747
2.260
0.382
0.830
0.369
1.060
S.W.
-0.638
-2.360
-0.443
-1.290
0.116
0.290
-0.209
-0.380
0.109
0.140
West
-0.807
-3.690
-0.066
-0.260
0.178
0.460
-0.600
-1.350
-0.364
-0.590
NE*ratings
0.104
0.350
-0.572
-1.590
-0.107
-0.260
0.578
1.010
-0.120
-0.210
South*ratings
0.506
1.570
0.794
2.290
0.048
0.110
0.349
0.680
-0.169
-0.320
West*ratings
0.943
2.700
0.298
0.660
0.016
0.030
1.037
1.760
0.010
0.010
SW*ratings
0.195
0.520
0.103
0.240
-0.181
-0.350
0.308
0.490
-0.446
-0.520
number of auctions
number of bids
'R-square'
'F-statistic'
'p-value'
1075
4300
0.29
17.24
0.00
1007
5035
0.48
30.24
0.00
852
5112
0.44
26.26
0.00
687
4809
0.43
25.47
0.00
531
4248
0.35
15.73
0.00
Table 3(b) : Test of equal indices
WA Coupon Rate
4
5
6
7
8
4
-1.924
-3.336
-2.917
-2.184
5
6
7
-1.157
-0.849
-0.445
0.289
0.585
0.318
5
6
7
-0.219
-0.424
0.538
-0.263
0.733
0.816
5
6
7
-0.553
-0.290
0.427
0.246
0.866
0.647
5
6
7
0.853
-0.920
-0.750
0.030
-1.453
0.030
5
6
7
0.853
-0.920
-0.750
-1.779
-1.453
0.030
WA Maturity
4
5
6
7
8
4
0.767
0.575
0.167
1.171
Type
4
5
6
7
8
4
-1.746
-2.339
-1.992
-0.858
Par amount
4
5
6
7
8
4
-0.788
-0.064
-1.600
-1.336
Bank Qualified
4
5
6
7
8
4
0.945
0.732
-1.600
-1.336